In many situations of interest, the potential theoretic approach to metastability allows to derive sharp estimates for quantities characterizing the metastable behavior of a given system. In this framework, the average metastable times can be expressed through the capacity of corresponding metastable sets, and capacities can be estimated with the application of two different variational principles, providing upper and lower bounds. After recalling these basic concepts and techniques, I will describe a new method to couple the dynamics inside potentials wells. Under some general hypothesis, I will show that this yields sharp estimates on metastable times, pointwise on any metastable set. Our key example will the random field Curie-Weiss model.

In many real-world networks, such as the Internet and social networks, power-law degree sequences have been observed. This means that, when the graph is large, the proportion of vertices with degree $k$ is asymptotically proportional to $k^{-\tau}$, for some $\tau \geq 1$. These networks are often small worlds, which means that distances in these networks are small. We will study two random graph models, the configuration model and the preferential attachment model, which will have power-law degree sequences when the number of vertices tends to infinity. An overview is given of known results about distances in these graph models. Also some new results will be presented, among which a $\log \log$ lower bound on the diameter of preferential attachment graphs with $\tau > 2$.

We begin with a simple random walk on (Z/NZ)^d, the d-dimensional integer torus with side length N for d >= 3, stopped at a time of order N^d. For large N, the integer torus locally looks like Z^d. What about the random subset of sites visited by the random walk? In the first part of the talk, we will see that this set locally looks like a random interlacement, a model that has recently been introduced by Sznitman. In the second part of the talk, we will see similar convergence results for several discrete cylinders of the form G_N x Z, where G_N is a sequence of finite connected weighted graphs of diverging size.

First we shall discuss uniqueness of the martingale problem corresponding to a degenerate SDE which models catalytic branching networks. This work is an extension of a paper by Dawson and Perkins to arbitrary catalytic branching networks. Next, we investigate the long-term behaviour of a particular system of SDEs for $d \geq 2$ types, involving catalytic branching and mutation between types. It can be shown that the overall sum of masses converges to zero but does not hit zero in finite time a.s. Finally, results on the relative behaviour of types are given to obtain some insight on how the process approaches zero.

Shankar Bhamidi (University of California, Berkeley, USA), December 5, 9 and 16, 2008
Joint work with David Aldous, Steve Evans and Arnab Sen.

The idea of local neighborhoods of probabilistic discrete structures (such as random graphs) to the local neighborhood of limiting infinite ob jects has been known for a long time in the probability community and has proved to be remarkably effective in proving convergence results in many different situations.

Here we shall give a wide range of examples of the above methodology. In particular

1. We shall show how the above methodology can be used to tackle problem of flows through random networks, where we have a random network with nodes communicating via least cost paths to other nodes. We shall show in some models how the above methodology allows us to prove the convergence of the empirical distribution of edge flows exhibiting how macroscopic order emerges from microscopic rules. 2. We shall show how for a wide variety of random trees (uniform random trees, preferential attachment trees from a wide variety of attachment schemes etc), how the above methodology shows the convergence of the spectral distribution of the adjacency matrix to a limiting non random distribution function. 3. Time permitting we shall also show how how one can deduce convergence of the maximal matching for various families of random trees and what this means about the spectral mass at 0.

Martin Hutzenthaler, Research assistant in the Dutch-German DFG/NWO research group "Mathematics of Random Spatial Models from Physics and Biology", November 24, 2008

Convergence to the Virgin Island Model

each other in each colony. Of special interest is a model with N colonies and uniform migration, that is, the target colony is chosen randomly in each migration step. Initially N-1 colonies are empty. As N tends to infinity, this model converges to the Virgin Island Model in which every emigrant populates an empty colony. Furthermore it turns out that the Virgin Island Model is optimal for the total population size among all branching-migration models with identical parameters for branching and competition. This supports the intuition that if there is competition among individuals and resources are everywhere the same, then the best strategy for survival is to move to unpopulated colonies. In addition there is a fairly explicit criterion for survival versus extinction of the total mass of the Virgin Island process.

As part of the research program of the Honours Programme 2007-2008, we, as a group of 4 students, studied complex networks. Complex networks are all around and understanding their structure can help increase efficiency of real networks. We studied an adapted version of the preferential attachment model by Barabási-Albert in which the clustering can be varied and we showed that this adapted model still has a power-law degree sequence with power-law exponent that we determined. Clustering in the old model was significantly lower than in real networks but the adapted model shows promising results of higher clustering. Using data from Rijkswaterstaat we further investigate whether the Dutch road network has some characteristics of a complex network.

We study the limiting behaviour of the large sums of strongly correlated exponentials as the number of their summands and the effective dimension of the correlation structure simultaneously tend to infinity. We consider two types of such sums which are generated by two a priori very different Gaussian correlation structures. The first type is a sum of hierarchically correlated random variables which is based on the partition function of Derrida's generalised random energy model with external field. The second type is an infinitesimal sum of genuinely non-hierarchically strongly correlated random variables which is based on the partition function of the Sherrington-Kirkpatrick model with multidimensional spins. We consider the asymptotic behaviour (the thermodynamic limit) of these two sums on a logarithmic scale (i.e., at the level of free energy) and also at a more refined level of their fluctuations (i.e., at the level of weak limiting laws). Interestingly for the Sherrington-Kirkpatrick model with multidimensional spins, we find traces of a hierarchical organisation in the thermodynamic limit. This supports the conjectured in theoretical physics universal behaviour of the sums of such sort.

Given a graph , a proper vertex colouring of  is -frugal if no colour appears more than  times in any neighbourhood and is acyclic if each of the bipartite graphs consisting of the edges between any two colour classes is acyclic. For graphs of bounded maximum degree, Hind, Molloy and Reed studied proper -frugal colourings and Yuster studied acyclic proper 2-frugal colourings. In this paper, we expand and generalise this study. In particular, we consider vertex colourings that are not necessarily proper, and in this case, we find qualitative connections with colourings that are -improper — colourings in which the colour classes induce subgraphs of maximum degree at most  — for choices of  near
to .

We study invasion percolation in two dimensions. We compare properties of the origin's invaded region to those of (a) the critical percolation cluster of the origin and (b) the incipient infinite cluster. We use tools from near-critical percolation in our analysis. This is a joint work with Michael Damron and Balint Vagvolgyi.

Suitably conditioned Brownian motion can be used to study Hamiltonians of quantum particles under holonomic constraints. We give some examples and discuss some of the related problems.

Current models for complex networks mainly aim to reproduce a number of graph properties observed in real-world networks. On the other hand, experimental and heuristic treatments of real-life networks operate under the tacit assumption that the network is a visible manifestation of an underlying hidden reality. For example, it is commonly assumed that communities in a social network can be recognized as densely linked subgraphs, or that Web pages with many common neighbours contain related topics. Such assumptions apply that there is an a priori "community structure" or "relatedness measure" of the nodes, which is reflected bythe link structure of the graph. A common method to represent "relatedness" of objects is by an embedding in a metric space, so that related objects are placed close together, and communities are represented by clusters of points. In this talk, I will discuss graph models where the nodes correspond to points in space, and the stochastic process forming the graph is influenced by the position of the nodes in the space.

The work presented was done in collaboration with William Aiello, Anthony Bonato, Colin Cooper, and Pawel Pralat.

Milan Bradonjic, (UCLA and LANL, USA)
Joint work with Aric Hagberg and Allon G. Percus

Combinatorial and Numerical Analysis of Geographical Threshold Graphs

We analyze the structure of random graphs generated by the geographic threshold model. The model is a generalization of random geometric graphs. Nodes are distributed in space, and edges are assigned according to a threshold function involving the distance between nodes as well as randomly chosen node weights. We show how the degree distribution, percolation and connectivity transitions, diameter and clustering coefficient are related to the weight distribution and threshold values.

Frank Redig (Leiden University ), May 16, 2008
Based on joint work with J.R. Chazottes and P. Collet (Paris).

We show how, starting from concentration inequalities, one can quite straighforwardly obtain bounds for the speed of relaxation to equilibrium (in L^p) for interacting particle systems. We will also discuss the weak Poincare inequality and how it can be applied in the context of disordered systems and systems at low temperature.

In many real-life networks, power-laws have been observed for the degree sequence, that is, the fraction of vertices with degree $k$ falls of as an inverse power of $k$. Furthermore, many networks are highly clustered, meaning that there is a large number of triangles and other short cycles. I will describe a random graph model, based on so-called random intersection graphs, where both the degree distribution and the clustering can be controlled. If time permits I will also describe some results on how the outcome of an epidemic on such a graph is affected by the clustering.

Balint Virag (University of Toronto), April 25, 2008
This is joint work with B. Valko

Improving a prediction of Wigner, Dyson (1962) gave a formula for the asymptotic probability of large gaps between beta ensemble eigenvalues.

We prove a slightly modified version of Dyson's predictions. The proof relies on the Brownian carousel, a continuum limit of the corresponding random matrices.

Rob Waters (University of Bristol), April 14, 2008

The duplicity of zero-one matrices

A matrix with entries in {0,1} can be regarded as a matrix over the integers, or over the field GF(2). We use combinatorial methods to show how much of a difference this distinction can make to its rank, and consider some related questions.

The coupling of stochastic dynamics is a powerful and general probability technique. It is an essential feature of perfect simulation algorithms, which allow to sample exactly from the stationary distribution associated to a (discrete-time) Markov Chain. Order preserving (or monotone) couplings are fundamental to their practical effectiveness. The development of such algorithms by Propp and Wilson in 1996 and their application to the Ising model and other statistical mechanics models is famous. We introduce, formalise and characterise the notions of monotonicity and complete monotonicity for Markov processes with discrete and continuous-time parameter, taking values in a finite partially ordered set. Complete monotonicity is the one required for perfect simulation. We state that the two notions are not equivalent in general. However, we show that there are partially ordered sets for which monotonicity and complete monotonicity coincide. We present and compare results for discrete and continuous-time processes.

Milos Stojakovic (University Novi Sad), march 4, 2008|
Joint work with Dan Hefetz, Michael Krivelevich and Tibor Szabo

For X a finite nonempty set and F a collection of subsets of X, the pair (X, F) is called a positional game on X. It is played by two players Maker and Breaker, where in each move Maker claims one previously unclaimed element of X and then Breaker claims one previously unclaimed element of X. Maker wins if he claims all the elements of a set in F, otherwise Breaker wins.

We study positional games on random graphs. Our main concern is to determine the threshold probability for the existence of Maker's strategy to win the game played on the edges of the random graph G(n,p), for various target families F of winning sets.

We consider random walk on mildly random environment on finite transitive d-regular graphs of increasing girth. After centering and scaling, the analytic spectrum of the transition matrix converges in distribution to a Gaussian noise. An interesting phenomenon occurs at d=2: as the limiting object changes from a regular tree to the integers, the noise becomes localized. The talk is based on joint work with Balint Virag.

Mandelbrot's fractal percolation process generates random fractal sets by an iterative procedure which starts by dividing the unit cube [0,1]^d in N^d subcubes, and independently retaining or discarding each subcube with probability p or 1-p respectively. This step is then repeated within the retained subcubes at all scales. As p is varied, there is a percolation phase transition in terms of paths for all d greater than or equal to 2, and in terms of (d-1)-dimensional "sheets" for all d greater than or equal to 3. After introducing the model and some known results, I will present new results concerning the discontinuity of crossing probabilities in all dimensions larger than 2, and the asymptotic behavior of critical values in dimension 2. (Joint work with Erik Broman.)

Given a finite set of lattice points A in Z^d, run a random walk started at the origin until it reaches a point adjacent to the complement of A, and remove that point from A. The internal erosion of A is the random set obtained by iterating this procedure until the origin itself is removed from A. I will present some results about internal erosion in one dimension, and conjectures in higher dimensions.

In recent joint work with Yuval Peres, we have defined a "smash sum" operation which given bounded open sets A and B in R^d, produces a set A\oplus B whose volume is the sum of the volumes of A and B. This operation is intimately related to the scaling limit of internal DLA. After reviewing the construction of the smash sum, I will explain how a process like internal erosion can be used to solve inverse problems such as the following: given a domain A in R^d, find a domain B such that A = B \oplus B.

Alessandro Pellegrinotti (Università degli Studi Roma 3), November 27, 2007

Random walk in fluctuating random environment

A review of the results concerning random walk in fluctuating in time random environment is given. The results concern the validity of the central limit theorem and the time behaviour of correlations.

presentation

This is work in progress joint with Frank den Hollander and Gregory Maillard. We study the long-time behavior of the heat equation on $\mathbb Z^d$ driven by a catalytic random potential. The solution of this equation describes the evolution of a ``reactant'' under the influence of a ``catalyst''. In this talk the focus is on the case where the catalyst is modeled by a voter dynamics with opinions 0 and 1 and with a symmetric random walk transition kernel starting from either the Bernoulli measure or the equilibrium measure. We consider the annealed Lyapunov exponents, i.e. the exponential growth rates of the successive moments of the solution. We show that these exponents are trivial when the random walk is not strongly transient, but display an interesting dependence on the diffusion coefficent otherwise. The main obstacle is the nonreversibility of the voter dynamics, since this precludes the application of spectral techniques. Here the duality with coalescing random walks is the key to most of our analysis.

Francesco Caravenna (Universita` degli Studi di Padova), October 23, 2007
(Joint work with Erwin Bolthausen and Béatrice de Tilière)

We consider a model for a polymer interacting with an attractive wall through a random sequence of charges. We focus on the so-called diluted limit, when the charges are very rare but have strong intensity. In this regime, we determine the quenched critical point of the model, showing that it is different from the annealed one. The proof is based on a rigorous renormalization procedure. Applications of our results to the problem of a copolymer near a selective interface are discussed.

We consider the conditional large deviation behaviour of the sequence of words obtained by cutting an i.i.d. sequence of letters according to a renewal process when fixing a typical letter sequence, and discuss potential applications to stochastic systems in random media. This is joint work in progress with Andreas Greven and Frank den Hollander.

Michael Steele (Wharton University of Pennsylvania), Augsut 20, 2007

Probability Inequalities: Concentration via Local Changes

This talk will provide an introduction to rich and useful class of  probability inequalities the essence of which is that if you have a function Z of independent random variables that does not change "too much" if one of the arguments is replaced by an indepentdent copy then Z is itself "concentrated"--- e.g. Z has small variance or thin tails. We will also consider applications of these inequalities to random variables that are of interest in combinatorics and machine learning.

Julien Berestycki (Aix-Marseille 1 and Paris 6), August 14, 2007
Joint work with Nathanael Berestycki and Vlada Limic)

In this talk we prove an asymptotic formula for the number of blocks of a Lambda-coalescent at small times. This formula is a consequence of a more general connection that we obtain between the small-time behavior of Lambda-coalescents on the one hand, and branching processes and continuous random trees on the other hand. This point of view connects in a unified way several recent results of Bertoin and Le Gall, Birkner et al., and Berestycki et al.

Mixing times and diophantine approximations

We study a simple shuffle where the top card of a deck is moved either to the bottom or to position $k$. It turns out that the relaxation time for the location of a card depends in a non-trivial way on the diophantine approximations to k/n.

Invasion percolation is a dynamic process closely linked to critical percolation, but without an external parameter. In joint work with Omer Angel, Frank den Hollander and Gord Slade we showed that the cluster of invaded points consists of a backbone together with sub-critical trees hanging off the backbone, with parameters that increase towards criticality.
By this analysis we derive scaling estimates for the r-point functions and related quantities, and show that the scaling behaviour differs from the incipient infinite cluster. In ongoing joint work with Mathieu Merle, we also analyze the continuum limit of the random trees obtained in invasion percolation

Emilio Cirillo (Università di Roma "La Sapienza", Italy), June 12, 2007

Decay of correlations in disordered systems

Disordered systems are statistical mechanics models with interactions chosen at random with a given probability distribution. In high temperature finite range systems, in which the interactions can be arbitrarily large with finite probability, the problem of the Griffiths' phase is recovered. By using a multiscale cluster expansion it is possible to prove the exponential decay of correlations in the whole complete analiticity region of the unperturbed model.

Results on the semi-infinite Ising model

Using an extension of Pirogov-Sinai theory, which allow for a countable number of phases with interacting contours, the semi-infinite Ising at small temperature is presented as a many phase contour model. We determine the piecewise analytical structure of the surface free energy and the phase diagram is found. In the latter point we use the correlation inequalities of Lebowitz and Griffiths.
 

I will give a brief and non-rigorous introduction to lattice trees (a model for branched polymers in statistical physics), super-Brownian motion (a process whose value at time t is a [random] measure), the lace expansion (roughly speaking an inclusion-exclusion analysis of an indicator function) and the connection between them when d>8.

In this talk I will discuss a random graph model based on geometric preferential attachment.

In the course of the talk I will introduce this model by means of other preferential attachment models. We will mainly consider the power-law exponent  of the degree sequence, i.e. the fraction of nodes with degree  is proportional to .

Jozef Skokan (LSE and UIUC), May 14, 2007

Numbers in Ramsey Theory

Ramsey Theory reassures us that the complete disorder is impossible. In graph theory setting this means that for a given a graph $G$ and an integer $k>1$, in any coloring of the edges of the complete graph $K_N$ by $k$ colors, there exists a monochromatic copy of $G$ provided $N$ is large. The smallest integer $N$ with this property is called the Ramsey number $r_k(G)$.

In the first half of my talk, we will briefly survey the most interesting results when $k=2$. In the second half, we look at the case when $k>2$. Here we do not know much even if $G$ is a very simple graph, e.g., a cycle.
Abstract: Ramsey Theory reassures us that the complete disorder is impossible. In graph theory setting this means that for a given a graph $G$ and an integer $k>1$, in any coloring of the edges of the complete graph $K_N$ by $k$ colors, there exists a monochromatic copy of $G$ provided $N$ is large. The smallest integer $N$ with this property is called the Ramsey number $r_k(G)$.

In the first half of my talk, we will briefly survey the most interesting results when $k=2$. In the second half, we look at the case when $k>2$. Here we do not know much even if $G$ is a very simple graph, e.g., a cycle.

In this talk we will present some results that we obtained with Frank den Hollander about a model of copolymer in an emulsion. This model was introduced by F. den Hollander and S. Whittington in their paper “Diffusion of a heteropolymer in a multi interface medium”. We will focus on the super-critical case (when one of the two types of droplets percolates), and more particularly on the phase transition between full-delocalization in the infinite cluster and partial-localization at the interface between the infinite cluster and the other solvent. We will see that the order of the transition is exactly 2.

(Joint work with Frank den Hollander)

Louigi Addario-Berry (Oxford), April 18, 2007

A General Ballot Theorem

The classical ballot theorem states that for a symmetric simple random walk S_1,...,S_n, given an integer k with 0 < k < n, if k and n are of the same parity then P(S_i > 0 for all 0 < i < n | S_n=k) = k/n. We show that essentially the same result holds for any random walk S with mean-zero step size X in the range of attraction of the normal distribution, as long as k is O(sqrt(n)). We also show that this result is essentially best possible. This is joint work with Bruce Reed

Marco Lenci (Stevens Institute of Technology, USA), April 17, 2007

Recurrence for quenched-random Lorentz gases and similar systems

It is a safe---albeit imprecise---conjecture that most Lorentz gases in 2D are recurrent. We formalize this conjecture by means of a stochastic ensemble of Lorentz gases, in which i.i.d.\ random scatterers are placed in each cell of a co-compact lattice in the plane. We give results towards showing that recurrence is an almost sure property (topological typicality and a 0-1 law that holds in every dimension). The mathematical machinery, including an application of a beautiful theorem by Schmidt and Conze, can be pushed further in the case of simpler systems having the same structure. These include the so-called persistent random walks in random environment. For a large class of them we prove almost sure recurrence.

Dimitris Cheliotis (Bahen Center for Information Technology, Toronto, Canada), April 17, 2007
Joint work with Balint Virag

Repeating patterns for one-dimensional random walk in random environment. A law of the iterated logarithm.

We take a random walk (or diffusion) in a random one-dimensional environment, and we look at its graph at different, increasing scales natural for it. What are the patterns that we will see appear repeatedly? This is a classical problem in the theory of stochastic processes.
Surprisingly, despite the complexity of our process, there is a neat characterization of the repeating patterns.
The analogous result for random walk in a flat, deterministic environment is the well known functional law of the iterated logarithm of Strassen.

The first half of the talk will be introductory. We will describe the model and state some fundamental results for it.

Evsey Morozov (Institute of Applied Mathematical Research and Petrozavodsk University, Russia), April 17, 2007

Stability analysis of a multiserver system with a dependence between workload, input and service time

Including various dependencies in a queueing model to reflect real-life effects, makes it more realistic. Our study concerns a general multiserver queue allowing dependence of the service time of an arriving customer and next interarrival period on both the current waiting time and the server assigned to the arriving customer. There are essentially no assumptions on the (conditional) distributions describing the model. We combine the Markov property of the workload process (to describe dependencies in the framework of Lindley-type recursion) with the regenerative property of that process to study stability using a characterization of the limit behaviour of the renewal process of regenerations. For this system, we establish sufficient stability conditions which are close to being necessary.

Ross Kang (Oxford University), April 10, 2007

The t-improper chromatic number of random graphs

We consider the t-improper chromatic number of the Erdos-Renyi random graph. As usual, G(n,p) denotes a random graph with vertex set [n] = {1,...,n} in which the edges are included independently at random with probability p. The t-improper chromatic number chi^t(G) is the smallest number of colours needed in a colouring of the vertices in which each colour class induces a subgraph of maximum degree at most t. If t = 0, then this is the usual notion of proper colouring. When the edge probability p is constant, we show that chi^t(G(n,p)) is likely to be close to min{np/t, chi(G(n,p))}, as long as t(n) = o(log n) or t(n) = omega(log n). For the case t(n) = Theta(log n), we outline a conjecture for the asymptotic value of chi^t(G(n,p)) which is motivated by large deviations estimates. This is joint work with Colin McDiarmid.

Yuri Yakubovich, Universiteit Utrecht

Slicing Young diagrams of partitions and compositions

In this talk I will explain some connections between partitions and compositions and their limit shapes when their weight and length grow in a certain regime. More precisely, we consider a uniform measure on partitions of weight n and length m. For n and m growing to infinity with

m^3 = o(n) it was shown in 1940s by Erd"os and Lehner that there are

approximately m! times more compositions than partitions. It implies that many properties of the uniform measures are asymptotically the same for compositions and partitions growing in this regime.

I will show that certain properties of the uniform measures still coinside asymptotically in different grow regime, namely if m^2 = o(n). This will be explained via a notion of sliced Young diagrams.

Pierre Mathieu (Aix-Marseille I), March 27, 2007

Invariance principles for random walks with random i.i.d. conductances

I'll explain some aspects of the proof of the individual invariance principle for random walks on a percolation cluster from my paper with A. Piatnitski, regarding in particular the use of 2-scale convergence.

Then I'll mention more recent developments on invariance principles for random walks with random conductances.

Thierry Bodineau (Paris 7), March 27, 2007

Current large deviations in stochastic systems

Using the framework of the hydrodynamic limits, we will discuss the large deviations of the heat current through a diffusive system maintained off equilibrium by two heat baths at unequal temperatures. In particular, we will explain the occurence of a dynamical phase transition which may occur for some models.

Persi Diaconis (Stanford University), March 13, 2007

What do we know about the metropolis algorithm?

The Metropolis algorithm is a basic tool of scientific computing. Useful analysis of the algorithm lies far in the future. I will explain the algorithm through a cryptography application, show how it is characterized as an L1 projection, and present some examples where the running time can be estimated. This last uses techniques of Micro-local analysis and is joint work with Gilles Lebeau.

Wouter Kager (EURANDOM), February 20, 2007

Patterns on percolation clusters: ratios and limit theorems

Abstract For site percolation on the hypercubic lattice, a pattern is a prescribed configuration in a cube of fixed diameter. We show that such patterns occur with positive density on large percolation clusters, and that two distinct patterns must occur in a given ratio. These results are used to prove the ratio limit theorem for percolation, which states that the ratio of the probabilities that the cluster of the origin has sizes n+1 and n converges. For supercritical percolation, we obtain a slightly stronger result. Generalisations of these methods to other Markov random fields are also discussed.

Sara Brofferio (Université Paris-Sud), February 13, 2007

Some properties of the lamplighter's walks

Consider a lamplighter that randomly walks along a street and that randomly switches on or off the lampposts that stand at every crossing. In this way, one constructs a random process on the set of all possible configurations of 0 and 1 on the integer line (the lampposts) times the integer line it self (the position of the lamplighter). In the last years mathematicians have been interested in such stimulating model. I would like to presents two joint works with W.Woess, on this subject. We associate to the random walk of the lamplighter a random walk on a suitable graph, called Diestel-Leader graph, that is the horocyclic product of two trees. Thank to this approach, we obtain a deeper understanding of the geometry of the lamplighter walk and we are able to deduce the precise asymptotic behavior of the Green kernel and a complete description of the harmonic functions.

Pierluigi Contucci (Università Bologna), February 6, 2007
Joint work with Joel Lebowitz.

Correlation Inequalities for Spin Glasses

A correlation type inequality for spin systems with quenched symmetric random interactions is illustrated and proved. Monotonicity of the pressure with respect to the strength of the interaction for a class of spin glass models is derived. Consequences include existence of the thermodynamic limit for the pressure and bounds on the surface pressure. Conjectured inequalities are discussed.

Christof Kuelske (University of Groningen), February 5, 2007
 Joint with Arnaud Le Ny

Time-evolved mean-field measures and catastrophe theory

Consider the ordinary Curie-Weiss Ising model under the site-wise independent spin-flip dynamics which flips the spins symmetrically between plus and minus at constant rate one. This seemingly trivial system shows some peculiar transition phenomena as a function of time when one looks at it on the level of continuity of conditional probabilities (mean-field non-Gibbsianness.) The analysis of the problem is related to an analytical bifurcation phenomenon that can be treated to a great level of concreteness, arising e.g. in explicit parametrizations of critical curves in parameter place. Apart from the interest in the concrete problem we use this example to lay out some basic bifurcation theory ("catastrophe theory") that is useful for studies of phase transitions in mean-field models.

Francesco Guerra (Università Roma La Sapienza), February 5, 2007

Broken replica symmetry bounds in spin glass theory

We give a general description of the broken replica symmetry bounds for the free energy in spin glass theory. We show how to extend them to the case of many coupled replicas. In this frame, we discuss the phenomenon of spontaneous replica symmetry breaking, discovered by Giorgio Parisi.
Our presentation will be completely elementary and self-contained.

Alexis Gillett (VU Amsterdam), January 16, 2007

An introduction to renormalisation and discretisation techniques for continuum percolation models

Renormalisation and discretisation are two powerful techniques for studying spatial stochastic systems. In brief, renormalisation is analysing a model on a different scale and discretisation is when a grid is imposed on a continuous space. Although often conceptually simple, examples of these techniques in the literature tend to be difficult to read.

These techniques will be illustrated by examples from a new continuum percolation model. Continuum percolation is the study of the connected components of random graphs, where the vertices of the graph live on a continuous space. The talk should be accessible for a general mathematical audience, with a number of open problems being presented that should interest specialists in the field. This talk presents joint work with Misja Nuyens and is based on a recently submitted paper available at www.cs.vu.nl/~ajg/preprints.ph

Francesca Nardi (Technische Universiteit Eindhoven), Mini-course November, 28, December 5 and 19, 2006

Lecture of 28 November 2006 “ Metastability for Ising model with the Glauber dynamics” Abstract: we consider the ferromagnetic Ising model on finite volume, for large values of the parameter beta, the measure will be concentrated around the absolute minima of H. We consider Glauber dynamics. A very interesting problem is the study of the so called "metastable behavior", namely the mechanism which, starting from a local minimum of H (metastable state), leads the system, via a large deviation, to the state with minimum energy (stable state). This problem is important for studying theoretical models of metastability in statistical mechanics. From the probabilistic point of view, metastable decay is a first exit problem from a suitable domain in the configuration space, for a reversible Markov chain in which those transitions which increase the energy H are exponentially small in the parameter beta. In the limit beta going to infinite, this problem can be treated in great generality within the Freidlin and Wentzel theory of small random perturbations of dynamical systems (see[1]). We will compare the results of the pioneer papers of Neves and Schonmann (see [2] and [3]) with the more recent results obtained by Bovier and Manzo (see [4]) and Manzo Nardi Olivieri Scoppola (see [5]).

Lecture of 5 December 2006 “ Metastability for anisotropic Ising model with the Glauber dynamics” Abstract: we consider the anisotropic ferromagnetic Ising model on finite volume, for large values of the parameter beta evolving under Glauber dynamics. We study metastability in this framework for which the equilibrium Wulff shape is not a cube. Considering large finite volumes, small magnetic fields, and very low temperatures, we show that the typical paths in the transition from the metastable to the stable phase are through sequences of ‘non-Wulff’ configurations. These results are obtained by Koteck\'y and Olivieri in [6].

Lecture of 19 December 2006 “Metastability and nucleation for conservative dynamics.” We analyze metastability and nucleation in the context of a “local version” of the Kawasaki dynamics for the two-dimensional Ising lattice gas in the limit of low temperature and low density. We consider the local version of the model, where particles live on a finite box and are created, respectively, annihilated at the boundary of the box in a way that reflects an infinite gas reservoir. Let Lambda be a sufficiently large finite box in the two dimensional lattice. Particles perform simple exclusion on Lambda, but when they occupy neighboring sites they feel a binding energy -U < 0. Along each bond touching the boundary of Lambda from the outside, particles are created with rate rho=exp(-Delta beta) and are annihilated with rate 1, where beta is the inverse temperature and rho > 0 is an activity parameter. Thus, the boundary of Lambda plays the role of an infinite gas reservoir with density rho. We take Delta in the interval (U, 2U) where the totally empty (full) configuration can be naturally associated to metastability (stability). We are interested in how the system nucleates, i.e., how it reaches a full box when it starts from an empty box. We will compare the results by den Hollander, Olivieri and Scoppola in [7] using pathwise approach and results by Bovier, den Hollander and Nardi in [8] that combine geometric and potential theoretic arguments. A special feature of Kawasaki dynamics is that in the metastable regime particles move along the border of a droplet more rapidly than they arrive from the boundary of the box. The geometry of the critical droplet are highly sensitive to this motion.

[1] M. I. Freidlin and A. D. Wentzell, Random Perturbations of Dynamical Systems (Springer-Verlag, 1984). [2] E. J. Neves and R. H. Schonmann, Critical droplets and metastability for a Glauber dynamics at very low temperature, Comm. Math. Phys. 137:209–230 (1991). [3] E. J. Neves and R. H. Schonmann, Behaviour of droplets for a class of Glauber dynamics at very low temperatures, Probab. Theory Related Fields 91:331–354 (1992). [4] A. Bovier and F. Manzo, Metastability in Glauber dynamics in the low-temperature limit: beyond exponential asymptotics, J. Stat. Phys. 107:757–779 (2002). [5] F. Manzo, F.R. Nardi, E. Olivieri, E. Scoppola,``On the essential features of metastability: tunnelling time and critical configurations." Journal of Statistical Physics Vol. 115, (2004), 591-642. [6] Koteck\'y, R.; Olivieri, E. Stochastic models for nucleation and crystal growth. Probabilistic methods in mathematical physics (Siena, 1991), 264-275, World Sci. Publ., River Edge, NJ, 1992. 82C44. [7] F. den Hollander, E. Olivieri and E. Scoppola, Metastability and nucleation for conservative dynamics. Probabilistic techniques in equilibrium and nonequilibrium statistical physics. J. Math. Phys. Vol. 41, page 1424--1498 (2000). [8] A. Bovier, F. den Hollander, F.R. Nardi, ``Sharp asymptotics for Kawasaki dynamics on a finite box with open boundary." Probability Theory and Related Fields, Vol 135, 265-310 (2006).

Alex Gaudilliere (Roma 2), November 21, 2006

Random walks approximation for diluted gas under Kawasaki dynamics

We couple a lattice gas of labelled particles, with low density $\rho$ evolving under Kawasaki dynamics inside a large two-dimensional box, with a gas of of independent random walks.

We compare the trajectories of the particles in the two dynamics and show that, under some general hypotheses on the initial configuration and at inverse temperature not higher than $-cst\ln\rho$, the two dynamics essentially behave in the same way up to times much larger than $1/\rho$.

In particular we derive some estimates for the probability of seeing at time $t$ a set of given particles in a given set of sites which are similar to those obtained for independent random walks.

All these results cover the particular case of simple exclusion.

Gregory Maillard (EPFL Lausanne, Switzerland), November 23, 2006
Joint work with Roberto Fernandez

We are interested in the study of phase transitions for continuous, positive and non-stationary chains with complete connections. We use the Hierarchical model introduced by Dyson (1969--1972) in his seminal work on phase transition for the one-dimensional Ising model to build, in a very simple way, a one-sided version of a Hierarchical model for which the corresponding chain with complete connections does not satisfy uniqueness.

We will show why this result is complementary to those existing in the literature and discuss the optimality of uniqueness criteria.

This project concerns two aspects of the simulation of fracture. First analysis is done of the energy distribution within a lattice at different points during the fracture process by calculating several of its moments for different lattice sizes. The energy moments are hypothesized to exhibit multifractal scaling with the lattice size. Only for the point right before mechanical breakdown of the system, a deviation from monofractality is concluded.

Another aspect builds on the conclusion that random percolation can model strong-disorder fracture with respect to several properties that exhibit fractality. Analysis is done whether modifying random percolation into gradient percolation based on the damage profile from the fracture simulation can improve the similarity to strong-disorder fracture with respect to these properties. It is concluded that gradient percolation based on fracture's damage profile provides little improvement, but surprisingly, exaggerating this profile does improve the similarity.

We present a model for artificial cells and accompanying simulation results. Some important questions are raised with respect to possible analytical approaches to understand these results and the audience is invited to take active part in potential pathways to follow.

We introduce an interacting spin model to describe the output of a contact between two populations carrying different cultural attitudes. The problem of establishing the presence of abrupt swings is reduced to the study of the phase transitions of the model.

We discuss a relation between generalised random graphs (see e.g. Britton, Deijfen and Martin-Löf (to appear in J. Statist. Phys.)) and an  (Susceptible  Infectious  Removed/immune) epidemic model on a directed (possibly complete) network . In the epidemic model the vertex set  stands for the individuals and the edge set  stands for the set of connections between individuals.

Our generalised random graph model is described as follows: Every vertex (individual) has a 2-dimensional random vector  assigned to it all distributed as the random varioable . An edge from  to  is open with probability . and otherwise it is closed. Conditioned on the random vectors

We interpret this model in a epidemiological setting as follows: If there is an open edge from  to , then at least one infectious contact is made from  to . i.e. if  becomes infected itself, then  will be infected as well, if it has not been infected before.

We compare different “epidemics” with given expected  and and show that if  and  are independent, then the process with fixed  and  is a worst case scenario, in the sense that the probability of a large outbreak as well as the expected number of ultimately removed individuals is maximal.To obtain this result we use an idea of Kuulasmaa (J. Appl. Probab. 1982).  

Limiting shapes for deterministic internal growth models

We consider the problem of metastability for a stochastic dynamics with parallel updating rule and we study the exit from the metastable state evaluating the exit time and the typical exit path. Moreover we give sharp estimates on this exit time.

Recently there has been a lot of interest in the use of random graphs as models for complex networks. This has inspired a number of models for generating random graphs with prescribed degree distribution. A natural generalization of the problem of generating random graphs with given degree distribution is to consider spatial versions of the same problem, where geometric aspects play a role. I will describe results on Z.

Francesco Caravenna (Institute of Mathematics, University of Zurich), July 13, 2006
Joint work with J.-D. Deuschel

Pinning models with laplacian interactions in (1+1)-dimernsion

We consider a random field $\phi: N -> R$ with Laplacian interactions of the form $V(\Delta\phi)$, for a large class of potentials $V$, and with in addition a delta-pinning reward for the field to touch the x-axis, that plays the role of a defect line. Denoting by $\epsilon \ge 0$ the intensity of the pinning reward, we show that there is a phase transition at $\epsilon = \epsilon_c > 0$ between a delocalized regime $(\epsilon \le \epsilon_c)$, in which the field wanders away from the defect line, and a localized regime $(\epsilon > \epsilon_c)$, in which the field sticks close to it. Using an approach based on renewal theory we extract the scaling limits of the model. In particular, we show that in the critical regime $(\epsilon = \epsilon_c)$ the rescaled field converges in distribution toward the derivative of a symmetric stable Levy process of index 2/5.

Applications of probability theory to polymers have been numerous. Starting with Flory, who realised that polymers in good solvents can be described as self-avoiding random walks, probability theory has shown to be important in the investigation of a variety of problems in polymer physics. In this seminar I will present an analysis of so-called multilayer Markov chains and apply the results to a model of a tethered polymer chain in shear flow. It is found that the stationary probability measure in the direction of the flow is nonmonotonic, and has several maxima and minima for sufficiently high shear rates. This is in agreement with the experimental observation of "cyclic dynamics" for such polymer systems. Estimates for the stationary variance and expectation value were obtained and showed to be in accordance with our numerical results.

I will present a method introduced by Vdovichenko (1965) to compute the partition sum of the two-dimensional Ising model. The main idea of the method is to replace the sum over contour diagrams that represent the interfaces between the Ising spins by a sum over loop diagrams. This representation has the nice property that the contribution from all loop diagrams with s loops is just a product of s single-loop contributions. The single loops are generated by a transition matrix whose eigenvalues determine the single-loop contribution and thus the partition sum.

The fact that factorization into single-loop contributions takes place implies that to study a single Vdovichenko loop, we may ignore all the others. This raises the question whether one can use this approach to study properties of a single interface in the Ising model (e.g. the interface whose conjectured scaling limit at the critical point is SLE).

We present the following model: beads are threaded on a set of wires on the plane represented by parallel straight lines. We add the constraint that between two consecutive beads on a wire, there must be exactly one bead on each neighbouring wire.

We construct a one-parameter family of "uniform" Gibbs measures on the bead configurations. When endowed with one of these measures, this model present connections with the GUE ensemble. We explain then that this process is the limit of any dimer model on a planar bipartite graph when some weights degenerate.

The problem of entrance time in small sets is one of the oldest question in ergodic theory motivated by statistical mechanics (Boltzmann). We will review some of the results for dynamical systems, and the extensions to repetition times, statistics of extreme and quasi invariant measures. If time permits we will sketch a new proof of the exponential law based on an idea of Kolmogorov for proof of the CLT.

Exponential inequalities can be seen as rough large deviations upper bounds for observables which are not sums of random variables. Devroye inequalities are the corresponding variance estimates. We will consider the case of dependent random variables (non product measure) corresponding to dynamical systems and discuss the consequences for concentration and some applications. We will briefly sketch a proof based on a coupling argument.

Markus Heydenreich (EURANDOM/TUe), April 11, 2006
Joint work with Remco van der Hofstad

We investigate the scaling of the largest critical percolation cluster on a large d-dimensional torus, for nearest-neighbor percolation in high dimensions, or when d>6 for sufficient spread-out percolation.

We use a relatively simple coupling argument to show that, up to logarithmic corrections, this largest critical cluster scales like V^{2/3}, where V is the volume of the torus.

Interestingly, this is the same asymptotic behavior that Erds and Reni (1960) observed for the critical random graph, which is the special case of percolation on the complete graph.

Our results establish a conjecture by Aizenman from 1997, apart from logarithmic corrections. We discuss implications of these results on the dependence on boundary conditions for high-dimensional percolation. It turns out that scaling of the largest connected component under bulk boundary conditions, as studied by Aizenman (1997), is quite different from the scaling under periodic boundary conditions.

Our method makes crucial use of the results by Borgs, Chayes, van der Hofstad, Slade & Spencer (2005), where partial results of V^{2/3} scaling on high-dimensional tori were proved.

The report is devoted to the stochastic flows on R^d which consists of coalescing diffusion particles. The new type of the spatial stochastic integral with respect to such flow is introduced. Based on this construction the Girsanov theorem for the Arratia flow is proved. The Clark representation for the functionals from Arratia flow also is obtained. The investigation of the spatial properties of Arratia flow leads to the boundary value problems for the stochastic anticipating equations. The solutions to these problems are proposed.

Starting from the rigorous picture of the full scaling limit of critical site percolation on the triangular lattice, obtained in collaboration with C. M. Newman, I will discuss some mostly heuristic and conjectural new developments concerning the near-critical scaling limit of 2D percolation and related model (such as the minimal spanning tree) obtained in collaboration with L. R. Fontes and C. M. Newman. These include a type of conformal covariance that replaces the full conformal invariance typical of systems at the critical point.

The Internal Diffusion Limited Aggregation has been introduced by Diaconis and Fulton in 1991. It is a growth model defined on an infinite set and associated to a Markov chain on this set. We focus here on sets which are finitely generated groups with exponential growth. We present a shape theorem for the Internal DLA on such groups associated to symmetric random walks. For that purpose, we introduce a new distance associated to the Green function, which happens to have some interesting properties. In the case of homogeneous trees, we also get the right order for the fluctuations of that model around its limiting shape.

Stochastic networks consisting of a set of finite capacity sites where different classes of individuals move according to some routing policy are investigated. These networks are motivated by bandwidth allocation problems in wireless networks. The associated
(non-reversible) Markov jump processes are analyzed under a thermodynamic limit regime, i.e. with symmetry properties and when the number of nodes goes to infinity. Under some conditions on the parameters, a metastability property is proved. It is shown that, despite the fact that the dynamic of these networks is local, several equilibrium points coexist in the limit. The implications of this unusual property (for queueing networks) are discussed. The key ingredient of the proof of this result is a dimension reduction achieved by the introduction of two energy functions and a convenient mapping of their corresponding local minima and saddle points. Cases with a unique equilibrium point are also analyzed.

We introduce conditions that imply, in the multidimensional setting, a strong law of large numbers with non-vanishing limiting velocity (which we refer to as ballistic behavior) and an invariance principle with non-degenerate covariance matrix. As an application of our results, we give a concrete criterion on the drift term under which the diffusion in random environment exhibits ballistic behavior. This criterion provides new examples of ballistic diffusions. With our methods, we are further able to rederive the well-known ballistic character of a class of diffusions in random environment with a divergence-free drift by simply checking the above mentioned sufficient condition for ballistic behavior.

Jan Swart (UTIA, Academy of Sciences of the Czech Republic), December 16, 2005

The contact process seen from a typical infected site

In this talk, we will consider contact processes on general countable groups, which includes the usual d-dimensional integer lattice, regular trees, and more. We will see that the expected number of sites has a well-defined exponential growth rate, which is closely tied to how the process looks as seen from a typical (`Palmed') infected site at a typical late time. In particular, if the process grows subexponentially, as is for example the case on the usual integer lattice, we will prove that the process as seen from a typical infected site converges, as time tends to infinity, to the upper invariant measure conditioned on the origin being infected.

Jan Swart (UTIA, Academy of Sciences of the Czech Republic), December 9, 2005

Renormalization of catalyctic Wright-Fisher diffusions

Systems of linearly interacting diffusions indexed by the hierarchical group can be analyzed by means of a renormalization transformation, which tells one how the local laws defining the process are related to its behavior on large space and time scales. In the case where the single components are one-dimensional, this technique has been developed by Dawson, Greven, and others, who showed that the behavior of such systems is highly universal, in the sense that the behavior on large space and time scales is independent of the local laws. In the case where the single components are multi-dimensional, a much richer universality structure is expected, but here there are still a lot of open problems. In this talk I will give a general overview on renormalization of linearly interacting diffusions, with special attention to two-dimensional catalyctic Wright-Fisher diffusions, which were recently treated by Klaus Fleischmann and me.

Andrea Collevecchio (University D'Annunzio, Pescara, Italy), November 18, 2005

On the transience of processes defined on Galton-Watson trees

We introduce a simple technique to prove the transience of processes defined on trees generated by branching processes. We apply this method to once-reinforced random walk and vertex-reinforced jump process.

Pieter Trapman (Vrije Universiteit Amsterdam), November 11, 2005

Epidemics on networks

I will discuss a stochastic model, describing the spread of infection on (social) networks. The main properties of networks that has to be taken into account are the number of individuals, the distribution of the number of neighbors and the number of short loops in the network. The number of short loops is important because some of the contacts of infective individuals may be with individuals that are no longer susceptible. Short loops may arise naturally in social networks because of ``the friends of my friends are also my friends''.

In epidemic literature the deterministic model of pair approximations is proposed to analyse epidemics on networks. Because this model is deterministic it is not possible to use it for estimating the probability of extinction of the infection. I will spend a few words on pair approximations and give some drawbacks.

Another method proposed is approximating the network with random graphs. On these random graphs it is rather easy to describe the spread of an infection and to estimate the probability of extinction of the infection. However the proposed random graphs in literature do not contain small loops in it.

In this talk a construction of random graphs is explained, that do have the same degree distribution and the same number of loops of length three as the original network. By this construction the strong features of pair approximations and random graph methods are combined.

Akira Sakai (TU/E - EURANDOM), October 21, 2005

Lace expansion for the Ising model II. Bounds on diagrams

The lace expansion has been a powerful tool to investigate mean-field behavior for various stochastic-geometrical models. Recently, the lace expansion for the Ising model has been proved for the first time. In the previous talk, I explained what is the lace expansion for the Ising model, as well as its consequence assuming bounds on the expansion coefficients.

In this talk, we continue the derivation of the expansion. One of the key points for the derivation is the source-switching lemma, which was first discovered by Griffiths, Hurst and Sherman, and then developed by Aizenman. I will show how it is used to complete the expansion and how it is extended to prove diagrammatic bounds on the expansion coefficients. These diagrammatic bounds are optimal to prove the mean-field behavior above 4 dimensions (with sufficiently large coordination number).

Leandro P. R. Pimentel (EPFL, Laussane), September 29, 2005
This is a joint work with P. A. Ferrari and J. B. Martin

Roughening and inclination of competition interfaces

We study the \emph{competition interface} between two growing clusters for a simple growth model (\emph{last-passage percolation}) in a bidimensional sector. Using technology built up for geodesics in percolation and a relation with the totally asymmetric simple exclusion process, we show that the asymptotic inclination and the fluctuations of this interface depends on the geometry of the initial configuration.

Pierluigi Contucci (Universita di Bologna), September 16, 2005

The Ghirlanda-Guerra Identities

We will show a proof of the Ghirlanda-Guerra identities which only requires that the variance of the Hamiltonian grows like the volume (thermodynamic stability). Our result is expressed in terms of the model's covariance and applies to all Gaussian spin glass models.

Malwina Luczak (London School of Economics), September 14, 2005
Joint work with Svante Janson.

A simple solution to the $k$-core problem

We study the $k$-core of a random (multi)graph on $n$ vertices with a given degree sequence. We let $n \to \infty$. Then, under some regularity conditions on the degree sequence, we give conditions on its asymptotic shape that imply that with high probability the $k$-core is empty; and other conditions that imply that with highprobability the $k$-core is non-empty and the sizes of its vertex and edge sets satisfy a law of large numbers. Under suitable assumptions these are the only two possibilities. In particular, we recover the result by Pittel, Spencer and Wormald on existence and size of a $k$-core in $G(n,p)$ and $G(n,m)$.
Our method is based on the properties of empirical distributions of independent random variables, and leads to simple proofs.

I will show how to generate a finite range decomposition for the covariance of a gaussian measure on a lattice. And I will use such a decomposition to study the RG transformation for a generic polymer system with a weak perturbation to the gaussian measure.

The time-symmetric part of the stochastic action

The path space measure for a (non-equilibrium) stochastic process $P_{\Phi}$ may be formally related to another (equilibrium) pathspace measure $P_{0}$ through the "stochastic action"
\[
P_{\Phi} = \exp(\Delta\aleph_{\Phi,0}) P_{0}, \] where $\Delta\aleph_{\Phi,0}$ is the difference in stochastic actions between the two processes, and $\Phi$ is the so-called field that parametrizes the one process with respect to the other.

The (time-)antisymmetric part of this action is related to the entropy production and through it to the Galavotti-Cohen fluctuation relation.
The question naturally rises: Then what is the meaning and significance of the time-{\em symmetric} part?

Knowledge and understanding of the time-symmetric part is crucial for systems that are far from equilibrium (beyond the linear regime), such as biological systems. We introduce the notion of field-reversal (an analogue of time-reversal) in order to study its properties. With help of some examples and by seeking analogy with other results we will share some of the insight that we have recently obtained.

In this talk we consider long range percolation models on Z^d. For various families of non-summable edge probabilities (p_n), we show that there exists an integer K, such that the truncated model, in which all edges whose sizes is larger than K are suppressed has a infinite cluster almost surely.

Karl Petersen (University of North Carolina Chapel Hill, NC, USA), June 24, 2005

Factors Maps on Shifts of Finite Type and Measures

A one-step shift of finite type is among the simplest of dynamical systems to describe, and a one-block factor map is among the simplest models of information loss through coding. Yet this setup involves surprising complexity and leaves open many natural questions, some of which have practical implications for information handling. The difficulties begin to emerge when one realizes that the image of a one-step Markov measure is probably no longer Markov, and its entropy can be hard to determine. In recent work with Anthony Quas and Sujin Shin, we considered a relative Shannon-Parry property: does every ergodic measure on the image have a unique preimage of maximal entropy? The answer is no, but we showed that the number of maximal entropy lifts is at least finite. Key tools in this area are the ideas of compensation function (introduced by Boyle and Tuncel and developed by Walters) and relative pressure and equilibrium states (Ledrappier and Walters). In a paper with Shin that is in press, we compared two natural definitions of relative pressure and showed that they are almost equivalent. Related results of Shin have implications for the identification of measures of maximal Hausdorff dimension for the restrictions of expanding maps on manifolds to compact invariant sets.

M. Deijfen (Free University of Amsterdam), May 17, 2005

Random graphs with prescribed degree distribution

Recently there has been a lot of attention on random graph models with an arbitrary prescribed degree distribution. In particular, models with power law degree distributions have been studied. In this talk, we will consider the problem of generating a random graph with a prescribed degree distribution under the extra restriction that the graph should be simple, that is, it cannot contain any self-loops or multiple edges between vertices. A number of possible algorithms will be described and they are all shown to give the correct degree distribution in the limit of large graph size. If time permits, a spatial version of the problem of generating random graphs with prescribed degree distribution will also be discussed.

Jorge Kurchan (ESPCI) April 6, 2005

Hidden symmetry of Kramers' equation and the problem of finding reaction paths

Kramers' equation can be extended in a way that reveals an underlying (super) symmetry. I will show how one can use this ideas as a basis for theoretical and practical methods for the study of reaction currents in phase-space (a central problem in Physical Chemistry). The same methods can be used to reveal separatrices and resonant tori in Hamiltonian systems.

Gerard Hooghiemstra (University of Technology Delft), April 5, 2005

Distances in random graphs with i.i.d.. degrees

Nikolaos Zygouras (ETH Zürich), February 22, 2005

A central Limit Theorem for a Randomly Driven Semilinear Parabolic Equation

We study the asymptotic behaviour of the probability that a Brownian motion in R^m ( m > 2 ), starting at x , hits a non polar compact set K before large time t .

Reinforced random walk (RRW) is a broad class of processes which jump between nearest neighbor vertices of graphs, and prefer visiting often visited ones over seldom visited ones. These processes can be used to model phenomena with a "nostalgic" component. We will describe the behaviour of RRWs on certain graphs, in particular trees.

In this talk we will consider a discrete random pinning model with an entropic repulsion. We will be particularly interested in the convergence at high temperature of this model toward a continuous one. This continuous model brings us important informations on the discrete one.

Markus Heydenreich (Eindhoven University of Technology), December 14, 2004

Implications of the triangle condition for percolation on the infinite lattice above the critical dimension 

Consider the percolation model on the infinite lattice Z^d with nearest-neighbor bonds or finite range spread-out bonds. It is conjectured that the percolation on this graphs obeys mean field behavior if the dimension d exceeds the critical value 6. This implies the existence of various critical exponents. By means of the lace expansion it has been shown that the triangle diagram is small in high dimensions for the nearest-neighbor model and the sufficient spread-out model. During the talk I will show how this result can be used to deduce the existence of critical exponents with mean-field values.

Glauco Valle da Silva (Ecole Polytechnique Federale de Lausanne), December 13, 2004

Hydrodynamics for Interacting Particle Systems with moving boundaries 

We describe the hydrodynamics for two models each one consisting of two Interacting Particle Systems on the semi-line coupled at their boundaries. The mouvement of the boundaries is related to some dissipative feature of the system. Therefore we have to work in the absence of equilibrium measures and in these cases the general methods to prove hydrodynamics cannot be applied directly.

Reda-Jurg Messikh (EURANDOM), December 7, 2004

The Wulff shape of the two dimensional Ising model in the vicinity of T_c

It is known that for fixed temperatures, T<T_c, a default of magnetization induces a Wulff crystal whose shape depends on the anisotropy of the lattice. We show that if the linear scale grows to infinity and the temperature simultaneously goes to the critical point then, in a certain regime, we still have a Wulff shape. In this situation the directional dependence vanishes and a circular droplet appears. We also give an application to image segmentation. By submitting the Ising model to a magnetic field that depends on the initial data of an image, we obtain a large deviation principle whose rate function is a simplified Mumford-Shah functional.

Gregory Maillard (EURANDOM), November 30, 2004

Chains with complete connections and Gibbs measures

We introduce a statistical mechanical formalism for the study of discrete-time stochastic processes (chains) with which we give: General properties of extremal chains, conditions for the uniqueness of the consistent chain and results on loss of memory and mixing properties for chains in the Dobrushin regime. We discuss the relationship between chains and one-dimensional Gibbs measures: We establish conditions for a chain to define a Gibbs measure and vice versa and we discuss the equivalence of uniqueness criteria for chains and fields.

Anne Fey (EURANDOM), November 23, 2004

Stable, but critical"; infinite sandpile diagnostics

A way of investigating a Self Organised Critical (SOC) model, is to find a related model with a critical phase transition, and to demonstrate that the dynamics of the SOC model steer towards this phase transition. An example is to compare the sandpile model with the activated random walkers model. The last model has a critical density of walkers. This relation has been investigated in one dimension by Meester and Quant. Fey and Redig compare the sandpile model in higher dimensions, on an infinite grid, with a similar model: an initial sandpile configuration, characterized by a density, is stabilised using the sandpile toppling rule. There should be a critical density, corresponding to the SOC state. We proved bounds for this critical density, and found some interesting features of the phase diagram in between, like metastability.

Thomas Schreiber (Nicolaus Copernicus University, Torun, Poland), November 11, 2004

Phase separation phenomenon for polygonal Markov fields in the plane

The polygonal Markov fields (PMFs) are random collections of polygonal contours in the plane, interacting by simple exclusion. They enjoy a number of interesting properties, including isometry invariance and exact solubility at certain temperatures as well as sharing some features of the two-dimensional Ising model (e.g. spontaneous magnetisation at low temperatures). The purpose of this talk is to present phase separation results for the PMFs in the Dobrushin-Kotecky-Schlosman (DKS) set-up. We show that under microcanonical constraint enforcing excess of dominated phase we observe a single large contour enclosing a disk-shaped region (Wulff crystal) of dominated phase, surrounded by an ocean of dominating phase. We also provide bounds on typical fluctuations of the phase-separating line. Our argument is based on a new graphical construction in the spirit of Fernandez, Ferrari & Garcia, which replaces usual cluster expansion techniques and seems particularly well suited for the geometric setting of the PMFs.

Antar Bandyopadhyay (Chalmers University of Technology, Sweden), November 2, 2004
Joint work with Professor David J. Aldous

Endogeny for Recursive Tree Processes

Cristian Giardiná (Dipartimento di Matematica, Università di Bologna, Bologna, Italy), October 28, 2004

Spin glasses are disordered systems which display many interesting properties when compared to conventional ferromagnetic models, both at equilibrium and dynamical level. In this talk I will introduce a description of these systems in terms of Gaussian random variables with a proper covariance matrix. This setting allows us to treat on the same grounds both mean-field models (SK,p-spin,REM,GREM) and finite-dimensional models (Edward-Anderson,Random Field). Two main issues will be considered: a) existence of thermodynamic limit; b) stochastic stability property of the quenched equilibrium state. This last property yields a set of identities in the infinite volume limit which are expressed as sum rules relating overlap distribution for different number of replicas. These identities are part of the RSB Parisi solution. We show that they are a subset of the Ghirlanda-Guerra identities and we also present a numerical study of the sum rules on systems of finite size. This enables us to gain some informations on different physical pictures (Droplet/RSB) wich have been proposed to describe the quenched equilibrium state of short-range models.

A communication system consisting of a Shannon random code over a noisy binary channel can be described as a disordered system in the language of statistical mechanics. The relevant communication quantities, capacity and error exponent, can then be obtained from the large deviation properties of the associated "free energy density" in the termodynamic limit. A large deviation principle for log-Laplace integrals allows the computation of the rate function. This leads to a new formulation of Shannon's theorem, extended to a broad range of "decoding temperatures".

Nina Gantert (Universität Karlsruhe), September 21, 2004
Joint work with Sebastian Mueller

The critical branching random walk is transient

We consider Branching Markov chains which describe a cloud of moving and branching particles. We define a notion of recurrence and transience for this cloud and give a complete classification under the assumption that particles do not die. In particular we show that a critical Branching Markov chain is always transient. Applying this result to random walks in random environment we show that the critical branching random walk in a typical random environment is transient.

This is a work in collaboration with C. Bahadoran, H. Guiol, K. Ravishankar. We consider attractive irreducible conservative particle systems on $\Z$, without necessarily nearest-neighbor jumps or explicit invariant measures. We prove that for such systems, the hydrodynamic limit under Euler time scaling exists and is given by the entropy solution to some scalar conservation law with Lipschitz-continuous flux. Our approach relies on (i) explicit construction of Riemann solutions without assuming convexity, which may lead to contact discontinuities and (ii) a general result which proves that the hydrodynamic limit for Riemann initial profiles implies the same for general initial profiles.

George Kordzakhia (University of Chicago), June 8, 2004
Joint work with Steven Lalley.

We explain the renormalization group for low temperature contour models that was invented by Bricmont and Kupiainen in their original setup, the nearest neighbor random field Ising model. To provide a pedagogical exposition we concentrate our presentation on the construction of the groundstate of the model, for fixed typical realizations of the random fields. Here all main problems caused by the disorder are already visible, but no expansions are needed.

Our analysis of the REM-like trap model is based on the knowledge of the eigenvalues and eigenvectors of the infinitesimal generator. The main tool is a complex integral representation of the averaged value of observables, allowing us to rederive the aging behaviour of the model from spectral information. The same methods are used in order to investigate the effect of different time-rescalings in a slightly modified version of the REM-like trap model (defined as random walk on a Poisson point process), which makes the relation to the real REM more suggestive.

Karel Netocny (EURANDOM), May 18 and 19, 2004

An introduction to perturbation techniques in statistical mechanics. Part I: Cluster expansions and Pirogov-Sinai theory

In this informal minicourse I will give a simple account on one of analytic methods heavily used in the rigorous statistical mechanics, which is based on convergent perturbation expansions.

I will argue that many problems including understanding both the high-temperature and low-temperature behavior of a class of spin lattice models, either classical or quantum, boil down to the analysis of a simple model of `polymers' (or particles, excitations,...) interacting only via a hard-core exclusion. In this model, both pressure and the correlation functions can be expressed in terms of a convergent series, the algebraic and analytic properties of which will be discussed.

One of the most exciting applications of the cluster expansions is the analysis of the lattice models in the regime of phase transitions, where the Gibbs measure is not unique. From the technical point of view, one wants to show that, in particular models, the coexistence of different ground states gives rise to the coexistence of Gibbs measures that are only small perturbations of the ground states. This needs a non-trivial generalization of the cluster expansions to a new class of geometrical models called Pirogov-Sinai models, where the polymers (called contours in this context) carry an additional information about the surrounding spin configuration.

In the rest of time, I might discuss some less standard approaches to the cluster expansions, serving as an introductory exercise before a more systematic discussion of multi-scale approaches next week.

Karoly Simon (Technical University of Budapest), May 12, 2004
Joint with M. Keane and B. solomyak

Consider a graph directed iterated function system (GIFS) on the line which consists of similarities. Assuming neither any separation conditions, nor any restrictions on the contractions, we compute the almost sure dimension of the attractor. Then we apply our result to give a partial answer to an open problem in the field of fractal image recognition concerning some self-affine graph directed attractors in space.

An estimate of Beurling states that if $K$ is a curve from $0$ to the unit circle in the complex plane, then the probability that a Brownian motion starting at $-\epsilon$ reaches the unit circle without hitting the curve is bounded above by $ c \, \epsilon^{1/2}$. This estimate is very useful in analysis of boundary behavior of conformal maps, especially for connected but rough boundaries. The corresponding estimate for simple random walk was first proved by Kesten. In a joint work with Greg Lawler we extend this estimate to random walks with zero mean, finite $(3+\delta)-$moment.

Consider an one dimensional diffusion in a random environment W, where W is a two sided Brownian motion path. It is known that this process has a sub diffusive behaviour and there is a process b depending only on the environment, having an explicit description, and giving with large probability a good approximation for the value of the diffusion at large times. We present a method for studying some path properties of the process b; we focus on one only application of it. This method puts into a mathematical framework a renormalization group analysis proposed in the non-rigorous physics paper of Le Dousal et al [1]. The tools we use are a path decomposition for W and the renewal theorem. Then we consider the case where the environment W is a spectrally one-sided stable process with index in (1,2] and we compute the one-dimensional distributions of the process b.

[1] P. Le Doussal, C. Monthus and D. Fisher. Random walkers in one-dimensional random environments: Exact renormalization group analysis. Physical Review E 59, 5.(1999), 4795-4840.

Stationarity of the Lagrangian velocity of a passive tracer in compressible environments; invariant measures and the Stokes drift

We present a 1-D random particle process with uniform local interaction which displays some form of non-ergodicity, similar to contact processes, but more unexpected. Particles, enumerated by integer numbers, interact at every step of the discrete time only with their nearest neighbors. Every particle has two states, called minus and plus. At every step two transformations occur. The first one turns evry minus into plus with probability $\beta$ independently from what happens at other places and thereby favours plus against minus. The second is ``impartial". Under its action whenever a plus is a left neighbor of a minus, both disappear with probability $\alpha$ independently of the presence and fate of other pairs of this sort. If $\beta$ is small enough compared to $\alpha^2$ and we start with all minuses, the minuses never die out.

We consider a finite system $\mathcal{F}=\left\{f_1,\dots,f_m\right\}$ of maps defined on a compact interval or on a half line. Assume that all elements of $\mathcal{F}$ are strictly increasing and convex (but not necessarily strictly convex). We do not assume that all of them are contractive. Even, there may exist a common fix point at which some of them are expanding and none of them are contracting. The system $\mathcal{F}$ together with a vector $\mathbf{p}=\left(p_1,\dots,p_m\right)$ forms an random iterated function sytem (RIFS). We apply $f_i$ in each step independently with probability $p_i$. Under the condition that the RIFS is contracting on average in a rather wide sense, we prove that the Hausdorff dimension of any invariant measures (there are uncountably many in general) are less than or equal to entropy/Lyaponov exponent.

 We devised an efficient algorithm to calculate the mobility of charge carriers in a disordered medium, in which the Pauli Master equation for large systems of dimensionality 1-3 is solved exactly. For a constant density of states, the mobility shows Mott-type variable-range-hopping (VRH) behavior at low temperatures. We studied the contribution m(R,e) to the mobility from hops between sites with particular separations in distance R and energy e and found a scaling behavior of this function. We have set up a theory to account for this scaling behavior, from which we also obtain new microscopic insight into the VRH problem.

We will as well study the case of a Gaussian density of states, which has received very much interest in literature as well.

Finally some preliminary results for the AC mobility will be presented.

There is considerable evidence that for dimensions $d>8$, the scaling limit of lattice trees is super-Brownian motion. Proving convergence as a stochastic process requires proving convergence of the finite dimensional distributions and tightness. We discuss the main ideas of a proof of the first requirement for a sufficiently spread out model of lattice trees for $d>8$. The proof uses the lace expansion and induction arguments.

We start with a strictly stationary sequence of random variables produced from a measurable function on a probability space where an invertible measure preserving transformation is acting. If this (measure theoretical) dynamical system is represented as a factor of another such system (the latter is said to be an extension of the original one),then the stationary sequence can be redefined with essentially the same probabilistic properties on this extended probability space. However, passing to a new extended dynamical system sometimes might be useful in some respects. For example, this extended space may be furnished with some structure (like a system of sigma-fields) covenient for proving limit theorems for the stationary sequence in question. Note, that we get a system of sigma-fields if the extending dynamical systems is formed by sample paths of a certain stationary stochastic process (we call such an extension probabilistic). Thus, first we will discuss what we need to construct by a dynamical system a new one whose sample paths are functions with values in the original probability space. The next problem is how to construct a factorization map, that is how to map these sample paths to points of the original space in a measure preserving and compatible to dynamics way (it is this map which makes one of dynamical system an extension of another one). Both these problems will be answered in terms of some appropriate Markov transition probabilities defined on the original probability space. A very natural way to solve the first problem is to consider a random perturbation of the original deterministic dynamics. Various constructions providing, under appropriate assumptions, the factorization map will be discussed. Then it will be explained how such transition probabilities can be constructed if our measure theoretical dynamical systems are produced from some classes of smooth dynamical system enjoying certain hyperbolicity. Finally, we need to relate our stationary sequence lifted to the extended space, with other objects on this space (like sigma-fields) to make some probabilistic conclusions. The approach is intended to be an alternative to symbolic dynamics but is not developed enough to substitute it at present. Drawbacks and advantages of this approach and open questions also will be discussed.

Gordon Slade (University of British Columbia), October 8, 15, 22, 29, November 5 and 12, 2003

The lace expansion and its applications 

Several superficially simple mathematical models, such as the self-avoiding walk and percolation, are paradigms for the study of critical phenomena in statistical mechanics. It remains a major challenge for mathematical physics and probability theory to obtain a mathematically rigorous understanding of the scaling theory of these models at criticality. Above their upper critical dimensions, the lace expansion has become a powerful tool for the analysis of critical scaling of the self-avoiding walk, lattice trees, lattice animals, and both oriented and non-oriented percolation. Results include proof of existence of critical exponents, with mean-field values, and construction of the scaling limit. For lattice trees and critical percolation, the scaling limit is described in terms of super-Brownian motion. The lectures will provide an introduction to the lace expansion and some of its applications. No previous exposure to the lace expansion will be assumed, and necessary background will be provided.

We introduce a statistical mechanical formalism for the study of discrete-time stochastic processes (chains) with which we give: General properties of extremal chains, new sufficient conditions for the uniqueness of the consistent chain and results on loss of memory and mixing properties for chains in the Dobrushin regime. We prove a construction theorem for specifications starting from single-site conditioning. We discuss the relationship between chains and one-dimensional Gibbs measures: We establish conditions for a chain to define a Gibbs measure and vice versa and we discuss the equivalence of uniqueness criteria for chains and fields.

Andras Telcs (Budapest University, Hongary), September 9, 2003

Short Type Asymptotic for fractal spaces

This talk presets estimates for the distribution of the exit time from balls and short time asymptotic for fractal spaces. The main motivation is twofold.  On one hand search for minimal conditions on the other hand recapture Varadhan's short time asymptotic for sub-diffusive spaces.

Barry D. Hughes, (University of Melbourne, Australia), September 9, 2003

Power-law distributions in natural and social systems

I discuss a simple explanation for the frequent occurrence of power-law tails in statistical distributions. It is shown that if any of a variety of stochastic processes with exponential growth in expectation are killed (or observed) randomly, the distribution of the killed or observed state will exhibit power-law behaviour in one or both tails. 

Four stochastic processes are considered, two with discrete state space (homogeneous birth-and-death process and Galton--Watson branching process) and two with continuous state space (geometric Brownian motion and discrete multiplicative process). It is shown how this simple mechanism can explain power-law tails in the distributions of the sizes of incomes, cities, internet files, biological taxa; and in surname, gene family and protein family frequencies. Data are presented illustrating the power-law behaviour. 

The killed Galton--Watson process is especially interesting mathematically, as the associated probability mass function has its asymptotic form modulated by log-periodic oscillations, a phenomenon previously encountered in several probabilistic contexts, and becoming more widely recognized in physics. 

I also discuss the extension of the ideas presented to the description of stochastically growing networks, where models with power-law distributions of vertex degree are of considerable recent interest. 

The presentation is based on joint work with W.J. Reed (University of Victoria, Canada), and with D.Y.C. Chan and A.S. Leong (University of Melbourne).

The purpose of the talk is to establish mixing properties and the uniqueness of infinite volume limit in the stationary regime for a class of so-called polygonal Gibbs-Markov random fields in the plane as introduced by Arak (1982), admitting a representation in terms of equilibrium evolution of one-dimensional particle systems tracing polygonal boundaries in two dimensional space-time. Our method is based on appropriate coupling techniques used to control the propagation of 'disagreement paths'. The disagreement paths may percolate, but their spatial density tends to zero in long time asymptotics, which enables us to conclude the required mixing and uniqueness results. We conclude our talk arguing that neither mixing nor uniqueness are guaranteed in non-stationary regimes, where pathological behaviour can be enforced under boundary conditions exhibiting unbounded particle density growth.

Localization-delocalization phenomena for random interfaces

In this second lecture, I will mostly concentrate on the Sherrington-Kirkpatrick model at high temperature or high external field. While from the physical point of view the system is regarded as trivial in this region of parameters, from the mathematical point of view the situation is very different. The study of this region is quite rewarding since, as I will show, here one can show that the Parisi theory actually gives the right prediction for the infinite volume limit. In this context, I will introduce our "quadratic replica coupling" method. Moreover, one can obtain more refined results, i.e., central limit theorems for the fluctuations of overlaps and free energy.

Mean field spin glass models were introduced, around 30 years ago, as approximations for realistic models describing disordered magnetic alloys. The analysis of these supposedly simple models turned out to be actually very challenging. By now, a widely accepted physical picture of their thermodynamic and dynamic behaviour exists (the so called Parisi theory of "replica symmetry breaking"). However, from the rigorous point of view many fundamental issues are still open. In the first lecture, I will first introduce the basic concepts and the motivation for these models. Then, I will explain the idea of the very simple "interpolation method", which proved to be very useful in this context. In particular, after giving a sketch of the theory developed by Parisi and of its physical implications, I will focus on Guerra's "broken replica symmetry bounds", which show why Parisi's solution is a natural candidate for the infinite volume free energy of the model.

I will introduce the Bak-tang-Wiesenfeld model of self-organized criticality, give some elements of the Dhar-formalism, and discuss recent results on existence of the dynamics on infinite graphs. I will also present some new open problems.

Random subgraphs of high-dimensional tori

We study random subgraphs of high-dimensional tori where bonds are occupied with probability p. Examples are the hypercube {0,1}^n or tori \{0,...r-1}^n for some n>6. We define the critical value p_c to be the value of p for which the expected cluster size of any fixed point attains the value V^{1/3}, where V is the volume of the graph. The main results show that p_c really is the critical threshold when we assume a simple random walk triangle condition. Such a triangle condition has appeared in the past in work of Aizenman and Newman, and Aizenman and Barsky, and implies mean-field behaviour. Indeed, both the expected cluster size and the largest cluster behave rather differently for p< p_c compared to p> p_c. We obtain detailed estimates on the asymptotics of both the largest and the expected cluster sizes on both sides of the critical threshold, as well as within the scaling window. The proofs mainly use methods from both mathematical physics, such as differential inequalities and the lace expansion. We believe the results have implications for percolation on Z^d and may provide easier and more flexible proofs for some of the known results obtained through the lace expansion. This is joint work with Christian Borgs, Jennifer Chayes, Gordon Slade and Joel Spencer.

Coupled map lattices are models of spatially extended dynamical systems. They are defined on a lattice space by the composition of local chaotic dynamics on each site and of a coupling between the sites. The competition between the local chaos and the coupling gives rise to various features, like spatio-temporal chaos, intermittency or occurence and spatially coherent patterns. We study the case of a weak coupling between expanding maps of the circle. We present in this case a Large Deviations Principle for the spatio-temporal empirical measure associated to the system. The rate function has a simple expression in the setup of thermodynamical formalism associated to the system. Our proof is direct : we use neither the coding by a Gibbs system, nor the previous results on uniqueness of the equilibrium (or SRB) measure. The main step is a ``Volume Lemma'' estimate on the size of Bowen balls under Lebesgue measure.

We obtain a formula for the asymptotic behaviour of the Dirichlet heat kernel for large time in terms of the survival probability of a Brownian motion, under the assumption that the latter decays subexponentially for large time. We also obtain a lower bound for the Dirichlet heat kernel for arbitrary open and connected sets in Euclidean space.

Abstract: For a large class of expanding maps of the interval, we prove that partial sums of Lipschitz obervables satisfy an almost sure central limit theorem (ASCLT). We provide a speed of convergence in the Kantorovich metric. Maxima of partial sums are also shown to obey an ASCLT. The key-tool is an exponential inequality recently obtained. Then we derive exact convergence rates for the supremum of moving averages of Lipschitz obervables (Erdös-Rényi type law).

In the presentation we will consider a collection of Bak-Sneppen like models and a number of recent results concerning the original model on the circle. The last one can be defined as follows. Each vertex of a regular polygon accommodate a random variable `state' (fitness) with value in (0,1). At the initial moment all the states are independent and U[0,1] distributed. We update the states in a Markov way. Every time step we choose a vertex with minimal state and give to this vertex and to its neighbors three new independent states with distribution U[0,1]. The principle open problems related to the model are to determine the limit distribution F when the number of vertices tends to infinity and to prove the power low of some characteristics. An important step to understanding of Bak-Sneppen model is to consider it as a sequence of avalanches from a given threshold in (0,1). We will introduce a number of critical thresholds and will show how their uniqueness determines the shape of the limit distribution F. 

In this talk I will explain an approach to metastability developped with M. Eckhoff, V. Gayrard and M. Klein in one of the simplest examples, the Curie-Weiss model. This is meant to serve as a preparation to the mini-course by Markus Klein. The intention is to highlight the key ideas of our appraoch in a transparent example.

We will discuss some quantitative aspects of hitting times and return times in the framework of various mixing stochastic processes or, equivalently, of more or less chaotic dynamical systems observed through a partition of their phase space. The basic result is Poincare recurrence theorem telling that almost every point comes back arbitrarily close to itself provided the measure is invariant or the process is stationary. Ergodicity implies that the first hitting time is almost surely finite and that the mean first recurrence time is finite (this is Kac's lemma). A natural question is to find some sufficient conditions under which moments of hitting and return times are finite. We will do so and provide some examples. Another way of studying hitting and return times is to compute their distributions when the probability of the set goes to zero, which is typically the case for cylinder sets. It is easy to see that one has to normalize hitting and return times in order to get a non-trivial limiting law. When the process mixes sufficiently fast, the asymptotic law is exponential. We will discuss some recent results, in particular when the process is weak Bernoulli. Outside this class, the exponential law seems to be not the only possible law and a very weak Bernoulli example will illustrate this.

Analytic semigroups and integral equations. 
Joint work with K. Homan and P. Clement 

Semigroup settings for integral and delay equations are abundant, however, many of them are based on perturbation of a translation semigroup, which has no smoothing effects. We treat a scalar integral equation with a completely monotonic kernel by an analytic semigroup. The machinery of analytic semigroups and interpolation spaces allows to recover the regularity results known from classical Laplace transform approaches. We expect that the state space setting will be helpful in treating stochastic integral equations, using recent theory on stochastic perturbations of semigroups.  

Wolfgang Desch (Universität Graz, Austria), September 12, 2002

Lipschitz continuity of the stop operator 
Joint work with J. Turi 

The stop operator is a basic building block for the theory of rate independent phenomena (hysteresis). Suppose a particle is trapped in a convex set K. A velocity u(t) is given. As long as the particle can move freely, its velocity v is the given function u. However, if the particle hits the boundary of K, v(t) is the vector closes to u(t) so that the particle stays inside K. The operator mapping u into v is called the stop operator. The continuity properties of this operator are useful technical tools in the theory of hysteresis. Lipschitz continuity holds in the spaces of continuous functions and integrable functions, if K is a polyhedron or has sufficiently smooth boundary.

Random copolymers have monomer sequences which are determined by some random process but are then fixed. As such they are an interesting example of quenched randomness. Even for simple models (eg Dyck paths) there are no complete solutions for these quenched random systems. They can often be solved in an annealed approximation but it is known that sometimes this approximation gives qualitatively incorrect results. Morita suggested a set of improved approximations which can be regarded as partial annealing treatments. This talk will describe the idea behind the Morita approximations and will discuss their application to two well-known problems: random copolymers adsorption at an impenetrable plane, and random copolymer localization at an interface between two immiscible solvents.

Simple random walks on various types of partially horizontally oriented regular lattices are considered. The horizontal orientations of the lattices can be of various types (deterministic or random) and depending on the nature of the orientation the asymptotic behaviour of the random walk is shown to be recurrent or transient. In particular, for randomly horizontally oriented lattices the random walk is almost surely transient. It is worth noticing that random walks on general graphs, beyond their fundamental interest in probability theory, they arise as simple models of physical systems. Since general directed graphs are associated with (not necessarily Abelian) $C^*$-algebras, and quantum mechanics is naturally formulated in terms of $C^*$-algebras, the study of random walk on oriented graphs is of some relevance to the rapidly growing field of quantum information and communication.

We prove that in the space of smooth area-preserving diffeomorphisms there exist open regions (seemingly quite large) where the so-called universal diffeomorphisms are dense (in the C-infinity topology). A diffeomorphism f is universal if the set of diffeomorphisms obtained from f by iterations and by smooth coordinate transformations is dense among all diffeomorphisms. Roughly speaking, these results show that understanding of dynamics of virtually any area-preserving diffeomorphism of a plane with chaotic behaviour is not simpler than understanding of dynamics of all area-preserving diffeomorphisms altogether.

In this talk we will discuss various applications of joinings in Ergodic Theory. In particular, their relation to the old unsolved problem of Rokhlin on multiple mixing. We shall also discuss "joining" proofs of theorems due to Furstenberg, Kalikow and Marcus.

Consider bond percolation on ${\bf Z}^d$. It is well known that for $p>p_c$ there exists (except for a set of $\omega$ with probability zero) a unique infinite cluster $C(\omega)$, which has positive density. I will discuss the behaviour of simple random walk on $C(\omega)$. The problem divides into two parts. The first is to use fairly well known properties of supercritical percolation to control the volume and spectral gap of balls in $C(\omega)$. The second is to use `heat kernel' methods, which have mainly been developed for very regular graphs in this situation, where there are small local irregularities. 

Random substitutions, fractal percolation and broadcasting on trees

Some properties of random substitutions have proven to be very difficult to analyze, such as connectivity of the limit set for fractal percolation (Mandelbrot percolation) and the reconstruction property for broadcasting on trees. In this talk we use multi-valued substitutions to define events that either imply or are implied by these properties. Our techniques to analyze random and multi-valued substitutions enable us to give new bounds on the critical value of fractal percolation. This work generalizes ideas of Chayes, Chayes and Durrett, Dekking and Meester, and White

The high temperature region of the SK model in external field is considered. By suitably coupling two replicas, one can show that replica symmetry holds in an explicitly determined region of parameters, which falls short of the Almeida-Thouless line, associated to replica symmetry breaking in Parisi's theory. In this region, one can also prove central limit theorems for physical quantities like overlaps and free energy. The main tools used are cavity equations and concentration of measure inequalities.

As a particular model of non-equilibrium statistical mechanics, we consider networks of interacting oscillators, driven at the boundary by heat baths at possibly different temperatures. We discuss a few natural questions related to this model: the uniqueness of a stationary measure, its Gibbsian nature in case of equal temperatures, and some positivity and strict positivity results for the entropy production.

Ellen Saada (CNRS. LNRS-Rouen, France), April 16, 17 and 18, 2002

Introduction to hydrodynamic limits, the case of symmetric simple exclusion

In this first talk, we will introduce the notion of hydrodynamic limits, and the necessary analytical tools on the elementary example of the symmetric simple exclusion.

Gradient models

In this second talk, we will present the one and two blocks estimates, and super-exponential estimates, that enable to deal with gradient models.

We will present this method for the one-dimensional totally asymmetric simple exclusion (when the solution of the hydrodynamic equation is smooth).

References: Kipnis, Landim

This course is an introduction to the area of infinite dimensional diffusion processes. We will use the theory of stochastic evolution equations as presented in the book "Stochastic Equations in Infinite Dimensions" by G. Da Prato and J. Zabczyk to study diffusions associated to some nonlinear stochastic partial differential equations. We will be mainly concerned with the ergodic properties of solutions and estimation of the spectral gap of the corresponding infinitesimal generator. 

We consider a family of interacting particle systems with two conserved quantities: particles perform random walk on the 1d integer lattice, driven by the negative gradient of their joint local time. The systems are naturally interpreted as deposition models. We derive hydrodynamic limit of these systems. Asymptotically we obtain the following systems of

 hyperbolic conservation laws: $$ \partial_t \rho + \partial_x(\rho u) = 0, \qquad \partial_t u + \partial_x(\rho + \gamma u^2) = 0, $$ where $\rho$ is the density of particles and $u$ (the velocity field) is the negative gradient of their joint local time (i.e. deposition), $\gamma$ is a constant depending on the microscopic laws. This is joint work (partly in progress) with Benedek Valko.

L. Erdos (Georgia Institute of Technology, Atlanta, USA), March 4, 2002

Long time evolution of random Schrodinger equations

We consider a single quantum particle moving in a weakly coupled random environment. We prove that in the kinetic scaling limit the quantum dynamics can be described by the classical linear Boltzmann equation as if the particle were a fluid subject to Markovian random collisions. However, the corresponding classical picture is misleading as it suggests that the limiting dynamics is the integrated Brownian motion. We will show how quantum mechanics explains this discrepancy.

Silke Rolles (EURANDOM), February 27, 2002

Reconstructing a random scenery observed with random errors along a random walk path

We show that an i.i.d. uniformly colored scenery on Z observed along a random walk path with bounded jumps can still be reconstructed if there are some errors in the observations. We assume the random walk is recurrent and can reach every point with positive probability. At time k, the random walker observes the color at her present location with probability 1-delta and an error Y(k) with probability delta. The errors Y(k), k\ge 0, are assumed to be stationary and ergodic and independent of scenery and random walk. If the number of colors is strictly larger than the number of possible jumps for the random walk and delta is sufficiently small, then almost all sceneries can be almost surely reconstructed up to translations and reflections.

This is joint work with Heinrich Matzinger.

Robert Dalang (Institut de mathématiques, Ecole Polytechnique Fédérale, 1015 Lausanne, Switzerland), Feb. 28, 2002

Some non-linear stochastic p.d.e.'s driven by correlated Gaussian noise

Abstract. There has been much work over the last few years to develop a theory for stochastic partial differential equations in spatial dimensions greater than one, in the case where the driving noise is spatially homogeneous. Most results concern equations that are first order in time, such as parabolic equations, and there are fewer results for equations that are second order in time, such as the wave equation. For linear equations, necessary and sufficient conditions on the covariance of the noise can be given which ensure the existence of a real-valued solution, and in many cases, these conditions can be shown to lead to existence results for non-linear equations. In this talk, we shall describe some results for the wave equation and noises that are not spatially homogeneous, but concentrated on a lower dimensional surface, such as a hyperplane or a sphere. This is joint work with O. Lévèque.

Michael Scheutzow (Technische Universität Berlin), February 19, 26 and March 5, 2002

Stochastic Flows

The aim of the talk is to explain the look-down construction for historical interacting Fisher-Wright diffusions. A single Fisher-Wright diffusion has been obtained as the weak diffusion limit of functionals of a series of particle models with a finite (but in the limit diverging) number of particles.
In the early 90ties Donnelly and Kurtz constructed then a particle system with countable many particles which randomly embeds the approximating particle systems, and hence allows to represent the Fisher-Wright diffusion as an a.s.-limit of a series of functionals of that so-called look-down system.

Stochastic evolutions of Gibbs measures had recently justify the physical relevance of non-Gibbsian measures: In [1], it is proved that the Gibbs property of a measure can be lost in the course of a Glauber dynamics. It is therefore natural to investigate the case of other dynamics, such as Kawasaki or Kawasaki + Glauber. In this talk (and in [2]), we consider general local stochastic dynamics of Gibbs measures and show that the Gibbs property is conserved for small times. For the sake of simplicity, we shall focus on the simple symetric exclusion process and describe how abstract cluster expansion techniques lead to this result. This is a joint work with Frank Redig. [1] Possible loss and recovery of Gibbsianness during the stochastic evolution of Gibbs measures. A.C.D. van Enter, R. Fernandez, F. den Hollander and F. Redig, Eurandom preprint, to appear in Com.Math.Phys.

Guillaume Bonnet (University of North Carolina), January 7, 2002, joint ISS-SN Seminar

The Burgers Superprocess

Spatio-temporal models in population dynamics and genetics have been studied for over 3 decades. In one simplified, and now classical, model, organisms disperse as a random walk and reproduce according to a branching process. To this "microscopic" description corresponds a "macroscopic" one that describes the evolution of the geographic distribution of large populations. The link between the two scales is given by the limit theory of a diffusion approximation. The limiting process is a measure valued process called super Brownian motion. It is of mathematical and modeling interest to generalize this basic model by introducing interactions between individuals, in either, or both, of the reproduction and displacement mechanisms. The model we introduce corresponds to a highly local interaction in the motion (individuals tend to leave crowded sites). At the macroscopic scale on which we focus, this leads to a nonlinear stochastic partial differential equation that combines a Burgers type nonlinearity with a driving noise of the superprocess kind. A related model in population genetic is also discussed.

We present a few aspect of the reconstruction in the case of two-colors with jumps.

Hermann Thorisson (Science Institute University of Iceland), November 19, 21 and 22, 2001

Coupling

Coupling means the joint construction of two or more random elements. The aim is usually to establish some distributional relation between the individual elements. But the aim can also be the reverse: to turn a distributional relation into a pointwise relation. Examples are theturning of stochastic domination into pointwise domination, weak convergence into pointwise convergence, and liminf convergence of densities into pointwise convergence where the random elements actually hit the limit. This deepens our understanding of the distributional relation itself, may enable us to establish previously hard-to-prove facts by simple pointwise arguments, and often leads to unexpected new results.This talk starts off with the above examples and then moves to stochastic processes (exact coupling, shift-coupling, and epsilon-couplings). Applications to Markov Processes, Regenerative Processes and in Palm Theory will be indicated. The view is then extended to random fields with applications in Palm Theory, and finally to random elements under a topological transformation group which opens up many new possibilities for applications: self-similarity, exchangeability, rotational invariance, the Lorentz and Poincaré transformations, ...

********************************************

Taboo Stationarity

In this talk we consider the taboo counterpart of stationarity. Stationarity is the characterizing property of any two-sided limit process obtained by shifting the time-origin of a one-sided process to the far future. Similarly, taboo stationarity is the characterizing property of any two-sided limit process obtained by shifting the origin of a one-sided process to the far future \it{under taboo}, that is, conditionally on the process not having entered a taboo region of its state space up to the new time-origin. This is, for instance, an appropriate model for a fish population that has lived a long time in an isolated lake, will eventually become extinct, but is still non-extinct at the time of observation. We present a basic but amazingly simple structural characterization of taboo stationary processes and then take a closer look at the structure in the regenerative case.

*******************************************

Point-Stationarity in $d$ Dimensions and Palm Theory

Point-stationarity formalizes the intuitive idea of a simple point process in $R^d$, $d Ñ 1$, for which the behaviour relative to a given \it{point of the process} is independent of the point selected as origin. Note that point-stationarity is different from stationarity. Stationarity means that the behaviour of the point process relative to any given nonrandom site is independent of the site selected as origin. For instance, when $d = 1$, the Poisson process with constant intensity is stationary since if the origin is moved from 0 to any other fixed site $t$ then we still have a Poisson process with the same intensity. But if a point is added at 0 then we obtain a non-stationary process since if the origin is moved from the point which we placed at 0 to any other fixed site $t$ then there need not even be a point there. This new process, however, is point-stationary since seen from the point which we placed at 0 the intervals between points are i.i.d. exponential and this will still be true if we move the origin, for instance, to the first point on the right of the point at 0. Thus for $d = 1$ point-stationarity has a straighforward definition: it means distributional invariance under shift of the origin to the $n$th point on the right (or left) of the point at 0. For $d > 1$ this definition clearly does not work. But is there some similar way of moving between points in higher dimensions? In this talk we define point-stationarity in $d > 1$ dimensions. Our definition turns out to be the characterizing property of the so-called Palm version of a stationary point process. We also sketch a modified Palm theory where the following shift-coupling result is central: the stationary and point-stationary processes can be represented as a single process seen from different origins.

Reference: Thorisson, H. (2000). Coupling, Stationarity, and Regeneration. Springer, NY.

Dr. Christof Külske (Weierstraß-Institut für Angewandte Analysis und Stochastik, Germany), October 30, 2001

Selfaveraging of random diffraction measures

We consider diffraction at random point scatterers on general discrete point sets in R^n, restricted to a finite volume.  We investigate the speed of convergence of the random scattering measures applied to an observable towards its mean, when the finite volume tends to infinity and prove an explicit universal large deviation upper bound. This shows the use of a simple cluster expansion in a situation that at first sight does not look like an interacting spin system.

Michiko Yuri (Japan), October 26, 2001

Phase transitions and slow decay of correlations

In this talk, for piecewise $ C^0$-invertible Markov systems first we clarify a property of potentials which may cause phase transition, subexponential instability and so slow decay of correlations. More specifically, we construct conformal measures $ \nu $ which are {\it weak Gibbs measures} for potentials $ {\phi} $ of {\it weak bounded variation} and then we show the existence of equilibrium states $\mu $ for $ \phi $ which are equivalent to the weak Gibbs measures $ \nu. $ Under the existence of certain periodic orbits, we can show both phase transition and subexponential instability with respect to $ \nu. $ 

Branching Brownian motion among obstacles: decay rate and optimal strategy for survival

We will study finer properties of the already discussed model where the BBM tries to survive among Poissonian hard obstacles.

Nonexpansive maps arise in a variety of applications. They can be used for instance, to model discrete time diffusion processes on a finite state space, and so called discrete event systems. (In the talk we will see several examples.) In many applications one is interested in the asymptotic behaviour of the iterates of such maps.

For maps that are nonexpansive in the 1-norm or in the sup-norm, it is known that the asymptotic behaviour of the iterates is periodic and moreover,

there exist an upper bound for the period that only depends on the dimension of the ambient space. It is an interesting problem to determine for a given dimension the possible periods. In the talk I will discuss some results concerning this problem.

Restoration of Gibbsianness in statistical mechanics.

After the detection of pathologies of the image of Gibbs measures under some Renormalization Group transformations, a program of restoration of Gibbsianness has been initiated in 1995 by R.L. Dobrushin. After a short presentation of Gibbs measures and RG pathologies, we present the existing restoration notions (weak Gibbsianness, almost and fractal quasilocality), describe some of their thermodynamic properties and discuss some possible extensions.

In an attempt to find a simple qualitative explanation of global glacial cycles physicists in the beginning of the 80ies started to study systems describing particles moving in multimodal periodically changing potential landscapes. Their mathematical description consists in nonlinear differential equations with weak periodic signals of period $T$, perturbed by white noise with intensity $\epsilon$. The system described by such an equation is in {\em stochastic resonance} if the intensity $\epsilon$ is tuned with the input period $T$ in an optimal way. In the meantime physics literature abounds with examples of systems showing stochastic resonance with origins in many areas of natural sciences.\\ In my talk, I will report on recent progress in a mathematically rigorous understanding of this phenomenon.

Christoph Bandt (Ernst-Moritz-Arndt-Universitaet in Greifswald, Germany) September 11, 2001

Ordinal time series analysis

In ordinary statistics, there are rank methods which deal with random variables on an ordinal scale, by comparing values only. Apart from the work of Marc Hallin, no such methods seem to exist for random processes or time series.We explain how Kendall'a tau can be used as an autocorrelation function, and how a permutation entropy can be defined as a complexity measure for time series. If the series is generated by a one-dimensional piecewise monotonic map, then the permutation entropy coincides with the Kolmogorov-Sinai entropy. This is joint work with Gerhard Keller from
Erlangen and Bernd Pompe from Greifswald.

Mini-course on polymer models

Course consists  four related but independent topics:

1. Self-avoiding walks.
2. Random knotting in lattice polygons.
3. Models of polymer adsorption.
4. Heteropolymers.

He will give an overview on each of these topics, covering both mathematics, motivation and applications. 

The abstracts of his lectures are:

1. This lecture is designed to review some of the results known rigorously about self-avoiding walks and related systems, emphasising methods that work in all dimensions. The lecture will also discuss some open questions.
2. The aim of this lecture is to discuss methods for proving rigorous results about knotting and linking in lattice polygons and related systems such as closed ribbons. Geometrical measures of entanglement complexity will also be covered.
3. Self-avoiding walks in a half-space, interacting with a confining plane, are a useful model of polymer adsorption. The lecture will discuss what is known rigorously and present some open questions. 
4. If the vertices of a self-avoiding walk are coloured at random, then the coloured walk can be used as a model of a random copolymer. Questions about self-averaging naturally arise, and this lecture will discuss methods for proving self-averaging in several models of this type.

The simulation-based inferential method called indirect inference was originally proposed for statistical models whose likelihood is difficult or even impossible to compute and/or to maximize. In this talk, indirect inference is proposed as a device to robustify the estimation and testing for models where this is not possible or difficult with classical techniques such as M-estimators. We investigate the local robustness properties of indirect inference and we derive the influence function of the indirect estimator, and the level and power influence functions of indirect tests. These tools are then used to design indirect inference procedures which are stable in the presence of small deviations from the assumed model. In particular we quantify the trade-off between bias correction via simulation and stability in the estimation of drift and volatility for diffusion models. Examples from time series, spatial statistics, and finance are used for illustration. Specifically, we discuss robust inference for autoregressive moving average (ARMA) models, simultaneous space-time autoregressions (STAR), and stochastic differential equations.

Mihyun Kang (Korea Advanced Institute of Science and Technology), April 25, 2001

A finite group with a probability distribution induces a directed graph. We regard an element as adjacent to another element if the group difference between them has a positive probability. This defines random walks on the group.
We consider directed graphs induced by two finite groups with probability distributions. We combine them in a way that they meet at one element. The random walks on the combined graph are defined by the transition probability based on the probability distributions on the groups.
We investigate the first hitting time, which is defined as the minimum number of steps that it takes to reach an element, starting from another element. We derive the explicit formulas of the generating functions of the first hitting times, by adapting P. Diaconis's formula based on discrete Fourier transformations or group representations of finite groups.
As examples we consider the simple random walks on several graphs, such as the figure of eight and dumbbells. The simple random walk is the nearest neighbor random walk where there is equal probability of moving from an element to an adjacent element.

Arnaud Le Ny (University of Rouen), April 6, 2001

Thermodynamic properties and almost quasilocality of some renormalized measures

We discuss possible extensions of the thermodynamic formalism for Gibbs measures, using recent results of Pfister on the large deviation properties of asymptotically decoupled measures. For the decimation of the low-temperature Ising model we prove the variational principle, and derive almost sure quasi-locality.

Nina Gantert, (Universitaet Karlsruhe) March 23, 2001

Random walk in random environment and random scenery

Random walk in random scenery was introduced in 1979 by H. Kesten and F. Spitzer. They consider the sum of  i.i.d. random variables along the path of a random walk. We give a short survey of their results. Random walk in random environment and random scenery is an analogous model where we consider the sum of i.i.d.\ random variables along the path of a random walk in random environment (RWRE).

For a one-dimensional, recurrent RWRE, we give some fluctuation results for this sum, among them a law of the iterated logarithm.

An important role is played by the self-intersection local time of the random walk. We extend a result of P. R\'ev\'esz for the local time of a recurrent RWRE to the case of a transient RWRE with zero speed.

Shannon-MacMillan theorems for random fields along curves and lower bounds for surface-order large deviations

Let $P$ be a random field over the two-dimensional lattice $\Z^2$ with finite state space. We introduce the notion of specific entropy $h_c(P)$ of the field along a curve $c$ as the limit of rescaled entropies along the lattice approximations of the blowups of $c.$ We prove a corresponding Shannon-MacMillan theorem. This allows us to represent $h_c(P)$ as a mixture of specific entropies along the tangent lines of $c.$ The proof is accomplished in three steps. Assuming only stationarity for $P,$ a Shannon-MacMillan theorem along lines is proved by using ergodic theorems for skew products. Ergodicity and mixing properties for skew products of an ergodic transformation with a semigroup of mixing transformation are discussed. In a second step, we assume a strong $0$-$1$ law for $P$ and extend the result to polygons. Finally, the specific entropy along a curve is obtained by approximation. As an application, we use the specific entropy along curves to refine Föllmer and Ort's lower bound for the large deviations of attractive Gibbs measures in the phase-transition regime.

Last updated 25/02/09