School and Workshop on Random Polymers and Related Topics
(14 - 25 May 2012)


~ Abstracts ~

 

Minimum of a branching random walk
Elie Aidekon, Université Paris 6, France


We consider the evolution of a particle system in dimension 1. At each integer time, particles make independent steps then split. We are interested in the position of the leftmost particle of this population. We show that this minimum, once recentered around its mean, converges in law. Our proof gives a description of the trajectory of the whole path of the leftmost particle as well. This is the analog of the well-known result of Bramson in the setting of the branching Brownian motion.

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The continuum directed random polymer and the intermediate disorder regime in dimension 1+1
Tom Alberts, California Institute of Technology, USA


In this talk I will describe recent joint work with Kostya Khanin and Jeremy Quastel for taking a scaling limit of the 1+1 dimensional directed polymer model to construct a continuous path in a continuum environment. The scaling takes place under what we call intermediate disorder, and acts on time, space, and the environment. We end up with a one-parameter family of random probability measures (indexed by the temperature parameter) that we call the continuum directed random polymer. As the temperature parameter varies the paths cross over from Brownian motion to what is conjectured to be a continuum limit of last passage percolation. This cross over is an inherent feature of the KPZ universality class.

Joint work with Kostya Khanin and Jeremy Quastel (Toronto).

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Fleming-Viot selects the mininal QSD: the Galton-Watson case
Amine Asselah, Universite Paris XII, France


Consider N particles moving independently, each one according to the dynamics of a subcritical branching process unless it hits 0, at which time it jumps to the position of one of the other particles chosen uniformly at random. We show that there exists a unique invariant measure for this process, and that the stationary empirical mea- sure converges, as N goes to infinity, to the minimal quasi-stationary measure of the branching process conditioned to survive. (Joint work with P.Ferrari, P.Groisman and M.Jonckheere).

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Scaling exponents in directed polymers in random environment
Antonio Auffinger, University of Chicago, USA


In a model of directed polymers in random environment, we place i.i.d. non-negative random variables on the nearest-neighbor edges of Z^d and we study the associated Gibbs measure on simple random walk paths. A long-standing conjecture gives a relation between two "scaling exponents": one describes the fluctuation of the point-to-point partition function and the other describes the typical maximum displacement of the random walk. This relation is sometimes referred to as the "KPZ scaling relation." I will discuss work I just completed with Michael Damron, in which we give appropriate definitions of these exponents and a proof of the relation.

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The polymer pinning model in correlated random environment
Quentin Berger, Ecole Normale Supérieure de Lyon, France


In recent years, the Harris criterion for the "polymer pinning model" with i.i.d. disorder has been to a large extent understood and mathematically justified. A next natural step is to look at the model with spatially correlated disorder. A naïve application of the so-called Weinrib-Halperin (WH) criterion predicts whether the Harris criterion will be modified or not, according to the decay rate of correlations and to the critical exponent of the non-disordered model. Somewhat surprisingly, for the pinning model one finds regimes where the WH criterion is violated. I will describe results on this issue by Fabio Toninelli and myself, and also by Fabio Toninelli and H. Lacoin.

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The extremal process of branching Brownian motion
Anton Bovier, University of Bonn, Germany


Branching Brownian motion is a classical process describing the evolution of a branching population of idividuals performing Brownian motion in space. The analyis of the spatial front of the population is a fascinating problem that has been studied for over 30 year, with seminal contributions by McKean, Bramson, Lalley and Sellke, to name a few. A profound feature is the connection to the F-KPP equation, a canonical reaction-diffusion equation. Here we look deeper into the edge of the process and construct the asymptotic extremal process of branching Brownian motion in the limit of large time. The limiting process is a Poisson cluster process, where the positions of the clusters is a Poisson process with random exponential density. The law of the individual clusters is characterized as branching Brownian motions conditioned to perform "unusually large displacements", and its existence is proved. The proof combines three main ingredients. First, the results of Bramson on the convergence of solutions of the Kolmogorov-Petrovsky-Piscounov equation with general initial conditions to standing waves. Second, the integral representations of such waves as first obtained by Lalley and Sellke in the case of Heaviside initial conditions. Third, a proper identification of the tail of the extremal process with an auxiliary process, which fully captures the large time asymptotics of the extremal process.
Joint work with Louis-Pierre Arguin and Nicola Kistler

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Random polymers and localization strategies
Francesco Caravenna, University of Milano-Bicocca, Italy


-General introduction to some important classes of random polymer models: pinning models, copolymer models, directed polymers with bulk disorder, parabolic Anderson model.

-Focus on pinning and copolymer models. Free energy and localization phase transition. The (generalized) homogeneous pinning model.

-Lower bound through "generalized rare stretch strategies". Smoothing effect of disorder.

-The phase diagram of pinning models. Relevance and irrelevance of disorder.

-The phase diagram of copolymer models. Improved lower bound on the critical line.

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Burgers equation with homogeneous random forcing
Eric Cator, Delft University of Technology, The Netherlands


In this talk I would like to show how ideas from Last Passage Percolation can be used to prove the existence of solutions for the 1d Burgers equation with homogeneous Poisson forcing. The randomly forced Burgers equation has received a lot of attention in recent years, but the existence of solutions has so far depended heavily on compactness assumptions. We were able to adapt methods used in LPP, where the existence of semi-infinite maximizing paths has been proved using methods introduced by Newman et al, and later extended by Cator and Pimentel, to paths that minimize the relevant Lagrangian action of the Burgers equation.

This is joined work with Yuri Bakhtin and Konstantin Khanin.

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Directed random polymers and Macdonald processes
Ivan Corwin, Microsoft Research New England and Massachusetts Institute of Technology, USA


The goal of the talk is to survey recent progress in understanding statistics of certain exactly solvable growth models, particle systems, directed polymers in one space dimension, and stochastic PDEs. A remarkable connection to representation theory and integrable systems is at the heart of Macdonald processes, which provide an overarching theory for this solvability. This is based off of joint work with Alexei Borodin.

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Uniqueness of random gradient states
Codina Cotar, Fields Institute, Canada


We consider two versions of random gradient models. In model A) the interface feels a bulk term of random fields while in model B) the disorder enters though the potential acting on the gradients itself. It is well known that without disorder there are no Gibbs measures in infinite volume in dimension d = 2, while there are gradient Gibbs measures describing an infinite-volume distribution for the increments of the field, as was shown by Funaki and Spohn. Van Enter and Kuelske proved that adding a disorder term as in model A) prohibits the existence of such gradient Gibbs measures for general interaction potentials in d = 2. Cotar and Kuelske proved the existence of shift-covariant gradient Gibbs measures for model A) when d\ge 3 and the expectation with respect to the disorder is zero, and for model B) when d\ge 2. In the current work, we prove uniqueness of shift-covariance gradient Gibbs measures with given tilt under the above assumptions.

(joint work with Christof Kuelske)

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Annealed Brownian motion in a heavy tailed Poissonian potentials
Ryoki Fukushima, Tokyo Institute of Technology, Japan


Consider the d-dimensional Brownian motion in a random potential defined by attaching a non-negative and polynomially decaying potential around Poisson points. We introduce a repulsive interaction between the Brownian path and the Poisson points by weighting the measure by the Feynman-Kac functional. Under the (annealed) weighted measure, it is shown that the Brownian motion tends to localize around the origin and the properly scaled process converges in law to a Ornstein-Uhlenbeck process.

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Large deviations for the log-gamma polymer
Nicos Georgiou, University of Utah, USA


We are discussing upper tail large deviations for the logarithm of the polymer partition function, when the environment is iid log-gamma distributed. The method of proof utilises a Burke type property and gives the exact rate function and the law of large numbers for the partition function. This, in turn, gives quenched LDPs for the polymer chain and endpoint.

This is joint work with Timo Seppalainen.

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On oscillatory critical amplitudes in hierarchical models
Giambattista Giacomin, Université Paris 7- Denis Diderot, France


It is rather well know in the physical community since about thirty years that the critical behavior of hierarchical models typically displays is not pure power law times a constant amplitude. In fact, the amplitude is a non-trivial multiplicatively period function. The aim of this talk is to present an up-to-date account of what is known about this phenomenon, that goes well beyond the framework of statistical mechanics hierarchical models. In line with the main topic of the workshop, I will mostly discuss the case of hierarchical pinning models.

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Random Polymers: an introduction
Frank den Hollander, Leiden University and European Institute for Statistics, Probability, Stochastic Operations Research and its Applications, The Netherlands


The goal of this lecture is to give an introduction to the mathematical theory of random polymers. The main focus will be on describing what random polymers are, how they can be modelled, what questions may be asked about them, and what are the key quantities from probability theory and statistical physics that are relevant for their study. Several examples of random polymer models will be presented, including models with disorder.

The lecture serves as an introduction to the mini-courses by Francesco Caravenna and Timo Seppalainen.

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A multi-scale refinement of the 2nd moment method
Nicola Kistler, University of Bonn, Germany


Derrida's Random Energy Model [REM] is a simple model for mean field spin glasses which has played a fundamental role in the understanding of the Parisi Theory. It can be completely analysed through the so-called 2nd moment method. The REM is also the representative of a large universality class of "trivial" models, those displaying 1-step replica symmetry breaking. However, a plain application of the 2nd moment method for these models typically falls short. I will present a refinement of the method which allows to overcome this difficulty in a number of cases. It is based on a multi-scale decomposition, and a coarse graining. The method is elementary and robust: I will try to make the case by means of some concrete examples.

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Directed polymers with boundaries and KPZ in equilibrium
Gregorio Moreno Flores, University of Wisconsin, USA


We show that the partition function of directed polymers (with a suitable boundary condition) converges, in the intermediate regime, to the Cole-Hopf solution of the KPZ equation in equilibrium.

Coupled with some fluctuation bounds for specific solvable models, this approach allows us to recover the cube root fluctuations for KPZ in equilibrium. We also discuss some partial results for more general initial conditions and some related questions.

This is a joint work with J. Quastel and D. Remenik, and with T. Seppalainen and B. Valko.

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Metastability of the contact process on NSW graphs
Jean-Christophe Mourrat, École Polytechnique Fédérale De Lausanne, Switzerland


We link th eproblem of metastability of the contact process on NSW graphs to very general survival properties of the contact process on arbitrary trees using very basic arguments.
Joint work with T. Mountford, D. Valesin and Q. Yao.

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Exactly solvable random polymers and their continuum scaling limits
Neil O'Connell, University of Warwick, UK


I will describe some recent developments on random polymer models which have an underlying integrable structure which makes them exactly solvable.
The continuum scaling limits of these models are related to the KPZ equation and have some remarkable properties. Parts of this talk will be based on joint work with Ivan Corwin, Timo Seppalainen, Jon Warren and Nikos Zygouras.

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A copolymer at a selective interface: variational characterization of free energy
Alex Opoku, Leiden University, The Netherlands


Consider a disordered polymer chain, consisting of different monomer types, located near a linear interface separating two media. The monomer types are randomly arranged along the chain and each monomer type has an affinity for one of the media. The model of interest has an energy function that rewards monomer-medium matches and penalizes mismatches.

It is well known that the annealed and the quenched free energies of the model exhibit a localization-delocalization phase transition with unique critical curves separating the resulting phases. Whiles there are explicit formulas for the annealed free energy and critical curve, no such formulas are known for the associated quenched quantities. In a recent work with Erwin Bolthausen and Frank den Hollander, variational formulas have been derived for these quantities. In this talk I will discuss these variational formulas and their consequences.

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A variational formula for the free energy of a copolymer in an emulsion
Nicolas Petrelis, Universite de Nantes, France


In this talk we consider a model for a random copolymer, made of monomers of type A and B, immersed in a micro-emulsion of random droplets of type A and B. The emulsion is modeled by large square blocks filled randomly with solvents A and B and the possible configurations of the copolymer are given by the trajectories of a partially directed random walk in dimension 2. The copolymer and the emulsion interact via an Hamiltonian which favors matches and disfavors mismatches.

In a similar model that has been studied recently, some restrictions were imposed to the random walk trajectories and a variational formula was derived for the quenched free energy per monomer.
In the present model we drop almost all those path restrictions and we display a new variational formula for the free energy. The latter formula involves auxiliary ingredients such as the free energies per columns of blocks and the frequency with which each type of column of blocks is visited.

The investigation of the phase diagram associated with this new variational formula is still in process but I will give some partial results about it.

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Self-duality of the last-passage percolation tree
Leandro Pimentel, Federal University of Rio de Janeiro, Brazil


The last-passage percolation tree is defined as the collection of semi-infinite coalescing geodesics. The aim of the talk will be to show self-duality of this tree.

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Fluctuations for the stationary 1D KPZ equation
Tomohiro Sasamoto, Chiba University, Japan


For the last few years, there have been renewed interests in the one-dimensional KPZ equation. In particular the height distributions of the interface have been computed for a few cases such as the narrow wedge initial condition[1].

In this talk we present explicit formulas of the height distribution and the two point correlation function for the KPZ equation in its stationary regime, which is one of the most natural and important case of the problem[2]. We explain basic ideas for the derivation based on replica method and discuss its importance from the point of view of statistical mechanics.

[1] H. Spohn, T. Sasamoto, PRL 104, 230602(2010)
[2] T. Imamura, T. Sasamoto, arXiv:1111.4634

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Directed polymers in random environments and KPZ universality
Timo Seppalainen, University of Wisconsin, USA


An introduction to the model of directed polymer in an i.i.d. random environment (bulk disorder). Phase transition between weak and strong disorder. Expected KPZ (Kardar-Parisi-Zhang) universality in 1+1 dimensions. Three exactly solvable models in 1+1 dimensions: the log-gamma polymer, the semidirected polymer, and the KPZ equation. The output theorem, or Burke property, in M/M/1 queues and polymer models, both in zero temperature and positive temperature. Fluctuation exponents for the log-gamma model.

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The one-dimensional KPZ equation and its universality class
Herbert Spohn, Technische Universität München, Germany


The KPZ equation is a stochastic PDE, which describes the motion of an interface bordering a stable phase against a metastable one. Over the last years we have seen interesting advances in terms of experimental realizations and in terms of exact solutions. We will outline both advances and the connection to directed polymers in a random potential.

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