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December 1216, 2016
"Guided Tour: Random Media"
Frank den Hollander
SUMMARY The aim of the proposed workshop is to bring together leading
researchers working on various aspects of research involving random
media and have them report on recent advances in their fields. In choosing the date we plan to hold our workshop close to Frank den
Hollander’s 60th birthday on December 1, 2016 and use the occasion to
celebrate his enormous contributions to the field.
ORGANISERS
INVITED SPEAKERS
MONDAY December 12 THEME: Polymers and Selfinteracting Random Walks
TUESDAY December 13 THEME: Random Walks in (static and dynamic) Random Environment
THEME: Dynamical GibbsnonGibbs Transitions and Path Large Deviations
THEME: Metastability
THEME: Parabolic Anderson Model and related topics
*************************************************************************************************************************************** Luca Avena Random walks in cooling random environments We propose a model of a onedimensional random walk in dynamic random
environment that interpolates between two classical settings: (I) the
random environment is resampled at every unit of time; (II) the random
environment is sampled at time zero only (i.e. a static random
environment). In our model the random environment is resampled along an
increasing sequence of deterministic times. We consider the annealed
version of the model, and look at three growth regimes for the
resampling times: (R1) linear; (R2) polynomial; (R3) exponential. We
prove laws of large numbers and central limit theorems. We list open
problems and conjecture the presence of a crossover for the scaling
behaviour in regimes (R2) and (R3). Ellen Baake Solving the recombination equation The recombination equation is a wellknown dynamical system from
mathematical population genetics, which describes the evolution of the
genetic composition of a population that evolves under recombination.
The genetic composition is described via a probability distribution (or
measure) on a space of sequences of finite length; and recombination is
the genetic mechanism in which two parent individuals are involved in
creating the mixed sequence of their offspring during sexual
reproduction. The model comes in a continuoustime and a discretetime
version; it can accommodate a variety of different mechanisms by which
the genetic material of the offspring is partitioned across its parents.
In all cases, the resulting equations are nonlinear and notoriously
difficult to solve. Elucidating the underlying structure and finding
solutions has been a challenge for nearly a century. Michiel van den Berg On Pólya’s inequality for torsional rigidity and first Dirichlet eigenvalue An inequality of Pólya asserts that for all open sets in Euclidean space with finite measure the product of torsional rigidity and first Dirichlet eigenvalue is bounded by its measure. We discuss the sharpness of this inequality and present some improvements for convex sets (joint work with Enzo Ferone, Carlo Nitsch and Cristina Trombetti). We also discuss bounds for the maximum of the torsion function in terms of the bottom of the spectrum of the Dirichlet Laplacian. Matthias Birkner Inhomogenous random walk pinning and conditional large deviations The inhomogenous random walk pinning model is a Gibbs transform of a random walk where intersections with another walk are rewarded/penalized with random weights. It is connected to various polymer models, in particular to directed polymers in random environment, via partial annealing, and undergoes a localization/delocalization transition in high dimensions. We discuss how partial annealing can be formulated in terms of conditional large deviation principles for the empirical process obtained by cutting an i.i.d. sequence along a renewal process and what this implies for the comparison of critical points. Erwin Bolthausen On twodimensional random walks in random environments We report about work in progress with Erich Baur and Ofer Zeitouni. For the standard model of random walks in random environments, dimension two is the critical dimension. This means that in leading order, the disorder stays constant after a renormalization procedure. However, fine analysis of the corrections to the leading order reveal that the disorder is indeed contracting. We give a detailed explanation of this effect. Anton Bovier Metastability according to Frank I will give a review on the developments in the theory of
metastability mostly from the perspective of the contributions made by
Frank, starting with his papers from Francesco Caravenna Universality in marginally relevant disordered systems We consider disordered systems of directed polymer type, for which
disorder is socalled marginally relevant. These include the usual
directed polymer in random environment in dimension (2+1), as well as
the disordered pinning model with tail exponent 1/2. We show that in a
suitable weak disorder and continuum limit, the partition functions of
such models converge to a universal limit. Connections with the
twodimensional Stochastic Heat Equation will be discussed. JeanDominique Deuschel Invariance principle for random walks with timedependent ergodic degenerate weights We study a continuoustime random walk, X, on Zd in an environment of
dynamic random conductances taking values in (0,∞). We assume that the
law of the conductances is ergodic with respect to spacetime shifts. We
prove a quenched invariance principle for the Markov process X under
some moment conditions on the environment. The key result on the
sublinearity of the corrector is obtained by Moser's iteration scheme.
Roberto Fernandez GibbsnonGibbs dynamical transitions: The largedeviation paradigm and how we got there The largedeviation paradigm has led to an alternative interpretation of GibbsnonGibbs transitions in terms of the number of optimal conditioned trajectories of evolving measures. This offers a new insight into the phenomenon that translates into the detection of further instances of these transitions. This talk will review the the definition and first examples of these transitions, contrast old and new approaches and present benchmark examples of the largedeviation paradigm. Cristian Giardina Dynamical properties of the inclusion process. Jesse Goodman Long paths in first passage percolation on the complete graph In a connected graph with random positive edge weights, pairs of vertices can be joined to obtain an a.s. unique path of minimal total weight. It is natural to ask about the typical total weight of such optimal paths, and about the number of edges they contain. To this end we consider the first passage percolation exploration process, which tracks the flow of fluid traveling across edges at unit speed and therefore discovers optimal paths in order of weight. On the complete graph, adding exponential edge weights results in optimal paths with logarithmically many edges  the same "small world" path lengths that are typical of many random graphs. However, by changing the edge weight distribution, we can obtain paths that are asymptotically longer than logarithmic. This talk will explain how tail properties of the edge weight distribution can be translated quite precisely into scaling properties of optimal paths. Andreas Greven Populations, genealogies and fluctuating environments We consider spatial population models and allow the coefficients of the dynamic to be random themselves, an old topic in probability. We come to a new point here. Typically for such evolutions we can define a genealogy even for models defined as solutions to SSDE. This we explain. Consecutively we point out that spatially homogeneous models can often be viewed as dynamic in a randomly fluctuating medium leading to genealogies evolving in randomly fluctuating medium, the medium given via the process of local population sizes. This is demonstrated with the example of interacting logistic Feller diffusions. Markus Heydenreich Random walk driven by interacting particle systems Some 10 years ago, Frank (dH) & Frank (Redig) started a research
program investigating random walk in dynamic random environment
involving Luca Avena, Florian Völlering, Renato dos Santos and many
others. The object of study are random walks whose transition rates
depend on an underlying random medium. The key point is then to let the
random medium evolve in time. This can be viewed as an interpolation
between classical RWRE models (where the medium is fixed in time) and
the homogeneous model. The challenge is to understand the impact of the
dynamic of the medium on the random walk characteristics. Wolfgang König The spatially discrete parabolic Anderson model with timedependent potential I will review Frank's works since 2006, jointly with Jürgen, Gregory,
and Dirk, on the longtime behaviour of the parabolic Anderson model on
the $d$dimensional lattice with timedependent random potential. Roman Kotecky Renormalization group in action Renormalization group transformations, when intrepreted as a flow in the space of Hamiltonians, are often leading to nonGibbs states. I will argue that properly formulated transformation on an extended space leads to useful applications. One example is the investigation of the strong convexity of the surface tension for random interfaces with nonconvex potentials. Christiaen Maes Pathlarge deviations approach to GENERIC
The talk reports on work in progress with Richard Kraaij, Alexandre
Lazarescu and Mark Peletier. The central question concerns the
structure of hydrodynamic and kinetic equations in describing the
relaxation to Gregory Maillard Parabolic Anderson model in a dynamic random environment: random conductances We consider a version of the Parabolic Anderson model where the
underlying random walk is driven by random conductances and investigate
the effect on the Lyapunov exponents. We will show that the annealed
Lyapunov exponents are controlled by pockets where the conductances are
close to the value that maximises the growth in the homogeneous setting
(constant conductances). In contrast, the quenched Lyapunov exponent is
controlled by a mixture of pockets where the conductances are nearly
constant. Julián Martinez Variational description of GibbsnonGibbs dynamical transitions for spinflip systems with a Kactype interaction We discuss the concept of Gibbs/ nonGibbs measure in the mean field
context and its extension to a localmean field model, and the emergence
of dynamical GibbsnonGibbs transitions under independent spinflip
("infinitetemperature") dynamics. We show that these dynamical
transitions are equivalent to bifurcations in the set of global minima
of the largedeviation rate function describing optimal conditioned
trajectories of the empirical density. Possible bifurcation scenarios
are fully determined in the mean field case, yielding a full
characterization of passages from Gibbs to nonGibbs and vice versa
with sharp transition times. Francesca Nardi Competing Metastable States for general rare transition dynamics: model independent results, applications to BlumeCapel model and Probabilistic Cellular Automata Metastability is a physical phenomenon ubiquitous in first order phase transitions. A fruitful mathematical way to approach this phenomenon is the study of rare transitions Markov chains. For Metropolis chains associated with statistical mechanics systems, this phenomenon has been described in an elegant way in terms of the energy landscape associated to the Hamiltonian of the system. We provide a similar description in the general rare transitions setup. Moreover, the study of systems with multiple (not necessarily degenerate) metastable states presents subtle difficulties from the mathematical point of view related to the variational problem that has to be solved in these cases. We give sufficient conditions to identify multiple metastable states. Since this analysis typically involves nontrivial technical issues, we give different conditions that can be chosen appropriately depending on the specific model under study. We show how these results can be used to attack the problem of multiple metastable states. Beside their theoretical content, our general results are a useful tool to approach metastability for the Blume–Capel model for a particular choice of the parameters for which the model has two multiple not degenerate in energy metastable states and for nonMetropolis systems such as Probabilistic Cellular Automata. In particular we estimate in probability and in expectation the time for the transition from any of the metastable states to the stable and we identify the set of critical configurations that represent the minimal gate for the transition. Nicolas Pétrélis Collapse transition of the 2dimensional selfinteracting prudent walk In this talk we will consider a model of self interacting prudent
walk in dimension 2. Elisabetta Scoppola Irreversible dynamics at low temperature: metastability and convergence to equilibrium Some results on irreversible dynamics are presented pointing out differences between reversible and irreversible dynamics. Gordon Slade Critical exponents for longrange O(n) models below the upper critical dimension We consider the critical behaviour of longrange O(n) models for n greater than or equal to 0. For n=1,2,3,..., these are ncomponent phi^4 spin models. For n=0, it is the weakly selfavoiding walk. For all n=0,1,2,..., we prove existence of critical exponents for the susceptibility and the specific heat, slightly below the upper critical dimension. This is a rigorous version of the epsilon expansion in physics. The proof is based on a rigorous renormalisation group method developed in previous work with Bauerschmidt and Brydges. Jan Swart Eigenmeasures and sharpness of the phase transition for the contact process It is well known that for contact processes, extinction is exponentially fast in the whole subcritical regime. In this talk, I will present a new and surprisingly short proof of this fact that is based on eigenmeasures, which are possibly infinite measures on the set of nonempty configurations that are preserved under the dynamics up to a timedependent exponential factor. In particular, in the subcritical regime, there is a onetoone correspondence between translationinvariant eigenmeasures and quasiinvariant laws of the process modulo translations. Rongfeng Sun Subdiffusivity of a random walk among a Poisson system of moving traps on Z We consider a random walk among a Poisson system of moving traps on Z. Previously, the quenched and annealed survival probabilities of this random walk have been investigated. Here we study the path of the random walk conditioned on survival up to time t in the annealed case and show that it is subdiffusive. As a byproduct, we obtain an upper bound on the number of socalled thin points of a onedimensional random walk, as well as a bound on the total volume of the holes in the random walk's range. Based on joint work with S. Athreya and A. Drewitz (to appear in Mathematical Physics, Analysis and Geometry). Jan Swart Eigenmeasures and sharpness of the phase transition for the contact process It is well known that for contact processes, extinction is exponentially fast in the whole subcritical regime. In this talk, I will present a new and surprisingly short proof of this fact that is based on eigenmeasures, which are possibly infinite measures on the set of nonempty configurations that are preserved under the dynamics up to a timedependent exponential factor. In particular, in the subcritical regime, there is a onetoone correspondence between translationinvariant eigenmeasures and quasiinvariant laws of the process modulo translations. Balint Toth Superdiffusivity of the periodic Lorentzgas in the BoltzmannGrad limit We prove central limit theorem and invariance principle under
superdiffusive scaling $\sqrt{t \log t}$ for the displacement of
particle in the $Z^d$based periodic Lorentz gas, in the BoltzmannGrad
limit. The result holds in any dimension. Stu Whittington Selfavoiding walks and related polymer models Selfavoiding walks are a standard model of the configurational
properties of long linear polymer molecules in dilute solution in good
solvents. They can be adapted to model polymer adsorption at a surface
and polymer collapse in a poor solvent. The vertices of the walk can be
coloured to model random copolymer phenomena such as localization at an
interface between two immiscible solvents. Using micromanipulation
techniques polymers can be extended, or pulled from one phase to
another, and selfavoiding walks are a useful model of such systems.
This talk will give a general survey of what is known rigorously about
these models, and will discuss several open questions.
PRACTICAL INFORMATION ● VenueEurandom, Mathematics and Computer Science Dept, TU Eindhoven, Den Dolech 2, 5612 AZ EINDHOVEN, The Netherlands
Eurandom is
located on the campus of Eindhoven
University of
Technology,
in the
Metaforum building
(4th floor) (about
the building). The university is
located at 10 minutes walking distance from Eindhoven main railway
station (take
the exit north side and walk towards the tall building on the right
with the
sign TU/e).
● RegistrationRegistration is free, but compulsory for speakers and participants. Speakers/Invitees please use this link to register: Registration speakers/invitees Participants please use this link to register: Registration participants
● AccommodationFor invited participants, we will take care of accommodation. Other attendees will have to make their own arrangements. For hotels around the university, please see: Hotels (please note: prices listed are "best available"). More hotel options can be found on the webpages of the Tourist Information Eindhoven, Postbus 7, 5600 AA Eindhoven.
● TravelFor those arriving by plane, there is a convenient direct train connection between Amsterdam Schiphol airport and Eindhoven. This trip will take about one and a half hour. For more detailed information, please consult the NS travel information pages or see Eurandom web page location. Many low cost carriers also fly to Eindhoven Airport. There is a bus connection to the Eindhoven central railway station from the airport. (Bus route number 401) For details on departure times consult http://www.9292ov.nl The University can be reached easily by car from the highways leading to Eindhoven (for details, see our route descriptions or consult our map with highway connections.
● Conference facilities : Conference room, Metaforum Building MF11&12The meetingroom is equipped with a data projector, an overhead projector, a projection screen and a blackboard. Please note that speakers and participants making an oral presentation are kindly requested to bring their own laptop or their presentation on a memory stick.
● Conference SecretariatUpon arrival, participants should register with the workshop officer, and collect their name badges. The workshop officer will be present for the duration of the conference, taking care of the administrative aspects and the daytoday running of the conference: registration, issuing certificates and receipts, etc.
● CancellationShould you need to cancel your participation, please contact Patty Koorn, the Workshop Officer.
● ContactMrs. Patty Koorn, Workshop Officer, Eurandom/TU Eindhoven, koorn@eurandom.tue.nl SPONSORSThe organisers acknowledge the financial support/sponsorship of:
Last updated
211216,

