April 24, 2014

February 14-18, 2011
Workshop on Interacting Processes in Random Environments

Talk Titles and Abstracts

On level sets of the local times, and application to polymers.
Amine Asselah
universite Paris-Est

We present estimates for the distribution of the size of level sets for a symmetric random in dimension 3 or more. We give application to estimating the annealed partition function for a class of charged polymer.


Burgers equation with Poissonian noise
Yuri Bakhtin
Georgia Tech

The Burgers equation is a nonlinear hydrodynamic model describing the evolution of the velocity field of sticky dust. The particles in this kind of medium interact only when they hit and stick to each other forming clumps. Some ergodic properties of this system with white-noise forcing and mostly in compact domains are known, but there are several interesting unanswered questions, especially for unbounded domains. In this talk a new simpler model for forcing based on Poissonian point field is proposed. The advantage of this model is that although it preserves many characteristic features of the white-noise model, it is easier to work with and visualize the resulting behavior. In fact, the model can be studied by looking at optimal paths through the Poissonian environment. In the unbounded domain case, if the spatial component of the measure driving the Poisson process has finite first moment, we obtain ergodic results for this model: one force-one solution principle; existence, uniqueness and some properties of a global skew-invariant solution including its behavior at infinity and a description of its basin of forward and pullback attraction; existence and uniqueness of a stationary distribution. Even for the Burgers equation on the circle this model provides a new insight into the behavior of the global minimizer.


A random walk pinning model, a conditional LDP and applications
Matthias Birkner
University Mainz
Coauthors: Andreas Greven (Erlangen) Frank den Hollander (Leiden) Rongfeng Sun (Singapore)

Consider a pair of transient random walks where the law of the second path is Gibbs transformed with a Hamiltonian proportional to the number of collisions with the first. The fact that here, the quenched and annealed critical points differ can be proved via a conditional LDP or via coarse-graining and fractional moment estimates. We discuss this result and its implications for the study of intermediate phases in certain interacting stochastic systems, in particular directed polymers in random environment and branching random walks in space-time random environment.


Non-ballistic random walks in random environments
Erwin Bolthausen
University of Zurich
Coauthors: Ofer Zeitouni

We outline the multi-scale approach for the analysis of the exit distribution of random walks in random environments in dimension three and above, developed with Ofer Zeitouni. We also report on work in progress in the critical two-dimensional case.


The weak coupling limit of disordered copolymer models
Francesco Caravenna
University of Milano-Bicocca
Coauthors: Giambattista Giacomin

A copolymer is a chain of repetitive units (monomers) that are almost identical, but they differ in their degree of affinity for certain solvents. A discrete model of such system, based on the simple symmetric random walk, was investigated in [Bolthausen and den Hollander, Ann. Probab. 1997], notably in the weak polymer-solvent coupling limit, where the convergence of the discrete model toward a continuum model, based on Brownian motion, was established. This result is remarkable because it strongly suggests a universal feature of copolymer models. In this talk we show that this is indeed the case. More precisely, we determine the weak coupling limit for a general class of discrete copolymer models based on renewal processes, obtaining as limits a one-parameter family of continuum models, based on stable regenerative sets.


Asymptotic speed of second class particles in a rarefaction fan
Eric Cator
Delft University of Technology
Coauthors: Leandro Pimentel and James Martin

In this talk we will consider one or more second class particles in a rarefaction fan in Hammersley's process or TASEP. We will show how we can use the concept of the Busemann function to characterize the joint distribution of the asymptotic speeds of the second class particles in an arbitrary deterministic initial configuration. We will show some examples for which we can actually calculate this distribution exactly.


KPP equation with random, time-space depedent, rate
Francis Comets
Univ. Paris Diderot
Coauthors: Errico Presutt

We consider the reaction-diffusion equation (du)(dt) = Du + k u(u-1) for D the Laplacian, t > 0, x ? Rd and u(0, .) (close to) the indicator function of the unit ball. The rate of reaction k > 0 is of the form k=k(t, x; w), it is stationary and ergodic, and, in fact, has a specific form. We prove existence of a limiting speed c, meaning that the solution at time t looks like the indicator function of a ball of radius ct + o(t). We also study the dependence of c on the fluctuations of k via a "disorder" parameter; a phase transition takes place, similar to the localization transition for polymers.


Probability Distribution of the Free Energy of the Continuum Directed Random Polymer in 1+1 dimensions
Ivan Corwin
Courant Institute, NYU
Coauthors: Gideon Amir, Jeremy Quastel

We consider the solution of the stochastic heat equation with multiplicative noise and delta function initial condition whose logarithm, with appropriate normalizations, is the free energy of the continuum directed polymer, or the solution of the Kardar-Parisi-Zhang equation with narrow wedge initial conditions. We prove explicit formulas for the one-dimensional marginal distributions -- the crossover distributions -- which interpolate between a standard Gaussian distribution (small time) and the GUE Tracy-Widom distribution (large time). The proof is via a rigorous steepest descent analysis of the Tracy-Widom formula for the asymmetric simple exclusion with anti-shock initial data, which is shown to converge to the continuum equations in an appropriate weakly asymmetric limit. The limit also describes the crossover behaviour between the symmetric and asymmetric exclusion processes.


CLT for biased random walk on multi-type Galton-Watson tree
Amir Dembo
Stanford University
Coauthors: Nike Sun

Let T be a rooted, multi-type Galton-Watson (MGW) tree of finitely many types with at least one offspring at each vertex and an offspring distribution with exponential tails. The r-biased random walk X(t) on T is the nearest neighbor random walk which, when at a vertex v with d(v) offspring, moves closer to the root with probability r/(r+d(v)) and to each of the offspring with probability 1/(r+d(v)). This walk is transient if and only if 0<r<R, with R the Perron-Frobenius eigenvalue for the (assumed) irreducible matrix of expected offspring numbers. Following the approach of Peres and Zeitouni (2008), we show that at the critical value r=R, for almost every T, the process |X(nt)|/sqrt(n) converges in law as n goes to infinity to a deterministic positive multiple of a reflected Brownian motion. Our proof is based on a new explicit description of a reversing measure for this walk from the point of view of the particle, a construction which extends to the reversing measure for a biased random walk with random environment
(RWRE) on MGW trees, again at a critical value of the bias.


A CLT for balanced, non-elliptic random walk in balanced random environment
Jean-Dominique Deuschel
Coauthors: Noam Berger

We consider a random walk on Z2 to nearest neighbors in a random environment. We prove a quenched invariance principle when the environment is i.i.d balanced and genuinely 2-dimensional, but not necessarily elliptic.


Large deviation rate functions for the partition function of directed polymers
Nicos Georgiou
University of Wisconsin-Madison
Co-author: Timo Seppalainen

We consider a 1+1 dimensional directed polymer model with certain log-gamma weights. This particular model allows for explicit calculations because of a Burke type -property. We present results about upper tail large deviations for the partition function, where the explicit rate function is readily computable utilizing the Burke property.


Random walk in random environment on trees
Yueyun Hu
Université Paris 13
Coauthors: Gabriel Faraud, Zhan Shi

For the random walk in random environment (RWRE) on Z, there are two well-known regimes: Sinai's slow movement in the recurrent case and Kesten-Kozlov-Spitzer's polynominal rate in the transient case. We discuss here a class of RWREs on trees and show that in the recurrent case, the walk may move slowly or may be subdiffusive according to the shape of some generating function. The asymptotic behavior of the RWRE on trees is closely related to that of some branching random walks.


Universality of KPZ equation
Milton Jara
Coauthors: Patricia Gonçalves

We prove under fairly general conditions that limit points of the density field of stationary, one-dimensional, weakly asymmetric, conservative processes are given by energy solutions of the KPZ equation. As an application we obtain the scaling limit of various functionals of such processes in terms of energy solutions of the KPZ equation.


Polymer measures and branching diffusions
Leonid Koralov
Department of Mathematics, University of Maryland

We study two problems related by a common set of techniques. In the first problem, we consider a model for the distribution of a long homopolymer in a potential field. For various values of the temperature, including those at or near the critical value, we consider the limiting behavior of the polymer when its size tends to infinity.

In the second problem, we investigate the long-time evolution of branching diffusion processes in inhomogeneous media. The qualitative behavior of the processes depends on the intensity of the branching. In the super-critical case, we describe the asymptotics of the number of particles in a given domain and describe the growth of the region containing the particles. In the sub-critical regime, we describe the limiting distribution of the total number of particles.


Exact critical behavior for pinning model in random correlated environment
Hubert Lacoin
CEREMADE Université Paris Dauphine
Coauthors: joint work with Quentin BERGER (Ens Lyon)

We investigate the effect of long range correlation in the environment for random pinning models. For a type of environment based on a renewal construction, we are able to describe the phase transition from the delocalized phase to the localized one, giving the critical exponent for the free-energy, and proving that at the critical point, the trajectories are fully delocalized. These results contrast both with what happens for the pure model or more studied case of i.i.d. disorder. where the critical behavior depends on Harris Criterion.


Directed polymers in random environment with heavy tails
Oren Louidor
Coauthors: Antonio Auffinger

We study the model of Directed Polymers in Random Environment in 1+1 dimensions, where the distribution at a site has a tail which decays regularly polynomially with power a, where a ? (0, 2). After proper scaling of temperature b-1, we show strong localization of the polymer to a favorable region in the environment where energy and entropy are best balanced. We prove that this region has a weak limit under linear scaling and identify the limiting distribution as an (a, b)-indexed family of measures on Lipschitz curves lying inside the 45-degrees-rotated square with unit diagonal. In particular, this shows order n transversal fluctuations of the polymer. If, and only if, a is small enough, we find that there exists a random critical temperature below which, but not above, the effect of the environment is macroscopic. The results carry over to d+1 dimensions for d > 1 with minor modifications. Joint work with Antonio Auffinger (NYU).


Directed polymers and the quantum Toda lattice
Neil O'Connell
University of Warwick

We characterise the law of the partition function of a Brownian directed polymer model in terms of a diffusion process associated with the quantum Toda lattice. The proof is via a multi-dimensional generalisation of a theorem of Matsumoto and Yor concerning exponential functionals of Brownian motion. It is based on a mapping which can be regarded as a geometric variant of the RSK correspondence.


Weak quenched limiting distributions for a one-dimensional random walk in a random environment.
Jonathon Peterson
Cornell University
Coauthors: Gennady Samorodnitsky

We study transient, one-dimensional random walks in a random environment (RWRE). A well known result of Kesten, Kozlov, and Spitzer gives the limiting distribution of such RWRE under the averaged probability measure (averaging over all environments). However, it was shown recently that for certain distributions on environments there may not be any quenched limiting distributions. That is, for a fixed environment (with probability 1) the random walk does not have a limiting distribution. In this talk I will describe recent work with Gennady Samorodnitsky that explains why (strong) quenched limiting distributions fail to exist. Viewing the quenched distribution of the hitting times as random probability measures, we show that they converge in distribution on the space of probability measures to a random probability measure with interesting stability properties.

Note: these results have also been obtained independently by Dolgopyat and Goldsheid and also Enriquez, Sabot, Tournier and Zindy.


Quenched Large Deviations for Random Walks in Random Environments and Random Potentials
Firas Rassoul-Agha
University of Utah
Coauthors: Timo Seppalainen and Atilla Yilmaz

We prove a process-level large deviation principle for quenched random walk in random environment subject to a random potential. In particular, both quenched random walk in random environment and quenched polymers in a random potential are covered. The walk lives on a square lattice of arbitrary dimension and has an arbitrary finite set of admissible steps. The restriction needed is on the moment of the logarithm of the transition probability and the potential, in relation to the degree of mixing of the ergodic environment. The rate function is an entropy and two variational formulas are given for the free energy.


Scaling limits of self repelling random walks and diffusions
Balint Toth
Institute of Mathematics, Budapest University of Technology
Coauthors: Illes Horvath, Pierre Tarres, Benedek Valko, Balint Veto

I will survey recent results on scaling limits of self-repelling random walks and diffusions which are pushed by the negative gradient of their own occupation time measure, towards domains less visited in the past. The typical examples are the so called 'true (or myopic) self-avoiding walk' or the 'self repelling Brownian polymer process'. It is proved that in three and more dimensions the processes scale diffusively, in two dimensions (this is the the critical dimension of the phenomenon) multiplicative logarithmic corrections are valid.


Percolation with a line of defects
Yvan Velenik
University of Geneva
Coauthors: Sacha Friedli and Dmitry Ioffe

We consider an inhomogeneous Bernoulli bond percolation process on the d-dimensional integer lattice (d>1). All edge occupation probabilities are given by p except for edges lying on the first coordinate axis which are occupied with probability p'. For any fixed p<p_c, we provide a detailed analysis of the consequences of the modified bond occupation probabilities p' on the exponential rate of decay of the connectivities along the line and on the behaviour of the corresponding cluster.


Charged polymers with attractive charges : a first order transition
Marc Wouts
Université Paris 13
Coauthors: Yueyun Hu (Université Paris 13) Davar Khoshnevisan (University of Utah)

We study a polymer model, which distribution depends on (quenched) random charges. Our main results are as follows. When the temperature is above a critical threshold, the distribution of the polymer converges (as the length of the polymer goes to infinity) to that of the random walk. Below the critical temperature, the maximum local time is of order the length of the chain. This transition is first order. In the low temperature regime, a large majority of the monomer lie on only four points, while the expectation of the end to end distance is bounded, uniformly in the length of the polymer.


Recursions, tightness and limit laws: from branching random walks to Gaussian free fields and FPP
Ofer Zeitouni
Weizmann Institute and University of Minnesota

I will describe several recent works (with Benjamini, Bolthausen, Bramson and Deuschel), concerning fluctuations of random fields. The common thread is an (approximate) underlying branching structure.


Strong Disorder in Semi-directed Random Polymers
Nikos Zygouras

Semi-directed, random polymers can be modeled by a simple random walk in a random potential. We identify situations where the annealed and quenched costs, that the polymer pays to perform long crossings are different. In these situations we show that the polymer exhibits localization.

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