Talk Titles and Abstracts
On level sets of the local times, and
application to polymers.
by
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.
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Burgers equation with Poissonian noise
by
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.
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A random walk pinning model, a conditional
LDP and applications
by
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.
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Non-ballistic random walks in random
environments
by
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.
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The weak coupling limit of disordered
copolymer models
by
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.
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Asymptotic speed of second class particles
in a rarefaction fan
by
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.
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KPP equation with random, time-space depedent,
rate
by
Francis Comets
Univ. Paris Diderot
Coauthors: Errico Presutti
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.
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Probability Distribution of the Free Energy
of the Continuum Directed Random Polymer in 1+1 dimensions
by
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.
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CLT for biased random walk on multi-type
Galton-Watson tree
by
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.
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A CLT for balanced, non-elliptic random
walk in balanced random environment
by
Jean-Dominique Deuschel
TU-Berlin
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.
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Large deviation rate functions for the partition
function of directed polymers
by
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.
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Random walk in random environment on trees
by
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.
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Universality of KPZ equation
by
Milton Jara
IMPA
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.
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Polymer measures and branching diffusions
by
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.
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Exact critical behavior for pinning model
in random correlated environment
by
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.
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Directed polymers in random environment
with heavy tails
by
Oren Louidor
UCLA
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).
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Directed polymers and the quantum Toda
lattice
by
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.
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Weak quenched limiting distributions
for a one-dimensional random walk in a random environment.
by
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.
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Quenched Large Deviations for Random Walks
in Random Environments and Random Potentials
by
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.
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Scaling limits of self repelling random walks
and diffusions
by
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.
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Percolation with a line of defects
by
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.
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Charged polymers with attractive charges
: a first order transition
by
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.
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Recursions, tightness and limit laws:
from branching random walks to Gaussian free fields and FPP
by
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.
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Strong Disorder in Semi-directed Random
Polymers
by
Nikos Zygouras
Warwick
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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