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ORGANISERS

Invited speakers

PROGRAMME YEP WORSKHOP

TUESDAY JANUARY 8

WEDNESDAY JANUARY 9

PROGRAMME SCHOOL RANDOM POLYMERS

THURSDAY 10-01

FRIDAY 11-01

SATURDAY 12-01

ABSTRACTS

Nick Beaton

Polymer adsorption on a rotated honeycomb lattice

In a recent paper by Bousquet-Mélou, de Gier, Duminil-Copin, Guttmann and myself, it was proved that a model of self-avoiding walks on the honeycomb lattice, interacting with an impenetrable surface, undergoes an adsorption phase transition when the surface fugacity is $1+\sqrt{2}$.

Our proof used a generalisation of an identity obtained by Duminil-Copin and Smirnov, and confirmed a conjecture of Batchelor and Yung. I consider a similar model of self-avoiding walk adsorption on the honeycomb lattice, but with the impenetrable surface placed at a right angle to the previous

orientation. For this model there also exists a conjecture for the critical surface fugacity, made in 1998 by Batchelor, Bennett-Wood and Owczarek. I prove that this is indeed the critical fugacity. Many of the arguments used are very similar to those featured in the previous paper, but the new

orientation also introduces a number of subtle complications.

Mathias Becker

Self-Intersection Local Times of Random Walks

Consider a random walk on the lattice $\Z^d$ whose steps have mean zero and finite variance. For $p>1$, we study the number $\|\ell_t\|_p^p$ of $p$-fold self-intersections up to time $t$. We derive the logarithmic asymptotics for $\P(\|\ell_t\|_p^p\geq r^p_t)$ for sequences $r^p_t$ in $(0,\infty)$, 
tending to infinity faster than $\E[\|\ell_t\|_p^p]$. The speed of the decay is identified in terms of mixed powers of $t$ and $r^p_t$, and the precise rate is characterized in terms of a variational formula. We will explain the appearance of two different regimes (sub- and supercritical case) 
and thus the two different strategies of the random walk to produce the required amount of self-intersections.

Quentin Berger

Influence of a correlated disorder in the polymer pinning model

Abstract: In the study of critical phenomena, the question of the influence of disorder is central. The discussion is whether the presence of randomness changes or not the critical properties of the system. This question of relevance/irrelevance of disorder has recently been a great 
source of

investigation in the polymer pinning model, and is now mathematically understood in the case of an IID disorder. After introducing the pinning model and describing the existing results, we will comment on the influence of spatially correlated disorder in this framework. In particular, 
we will stress how very strong correlations can

have a crucial impact on the critical behavior of the system. In particular, we show that a "strong disorder" regime appears, where large fluctuations of disorder make it relevant.

Francesco Caravenna (mini course)

Probabilistic aspects of polymers

The aim of this course is to give an overview on some probabilistic models that describe the behavior of an inhomogeneous polymer chain interacting with an environment. A key feature of these models is that they undergo phase transitions: a slight variation of some external parameters, such as the temperature, can have a huge impact on the large scale properties of the polymer, producing interesting localization phenomena. We will mainly focus on two classes of models, called pinning and copolymer models, for which substantial progress has been obtained in recent years. From a mathematical viewpoint, these models may be described as inhomogeneous perturbations of the law of a random walk, depending on the realization of an additional source of randomness (disordered systems). Some of the challenging problems that arise will be analyzed in some depth, using a range of probabilistic techniques.

Alessandra Cipriani
 
Fluctuations near the limit shape of Young diagrams under a conservative measure  
 
In this work we are considering the behavior of the limit shape of Young diagrams associated to random permutations on the set {1,...,n} under a particular class of multiplicative measures. Our method is based on generating functions and complex analysis (saddle point method). 
We show that fluctuations near a point behave like a normal random variable and that the joint fluctuations at different points of the limiting shape have an unexpected dependence structure. We will also give further directions of research concerned with the randomization of
the cycle counts of permutations and on the convergence to a continuous stochastic process. This is a joint work with Dirk Zeindler.

Alex Drewitz 

Effective polynomial ballisticity conditions for random walk in random environment

The conditions $(T)_\gamma,$ $\gamma \in (0,1),$ which have been introduced by Sznitman in 2002, have had a significant impact on research in random walk in random environment. They require the stretched exponential decay of certain slab exit probabilities for the random walk under the averaged measure and are asymptotic in nature. We show that in all relevant dimensions (i.e., $d \ge 2$), in order to establish the conditions $(T)_\gamma$, it is actually enough to check a corresponding condition $(\mathcal{P})$ of polynomial type on a finite  box. In particular, this extends the conjectured equivalence of the conditions $(T)_\gamma,$ $\gamma \in (0,1),$ to all relevant dimensions.
(joint work with N. Berger and A.F. Ramírez)

Gia Bao Nguyen

A variational formula for the free energy of a self-interacting partially directed random walk

Long linear polymers in dilute solutions are believed to undergo a collapse transition from a random coil (expand itself) to a compact ball (fold itself up) when the temperature is lowered, or the solvent quality deteriorates. A natural model for this phenomenon is a $1+1$ dimensional self-interacting and partially directed self-avoiding walk. In this paper,

we develop a new method to study the partition function of this model, from which we derive a variational formula for the free energy. This variational formula allows us to prove the existence of the collapse transition and to identify the critical temperature in a simple way. We also prove that the order of the collapse transition is $3/2$.

Alex Opoku

Entropy, Relative Entropy and more 

The notions of entropy, relative entropy and their densities will be reviewed in this course. We will exhibit :
• some of their key properties.
• how they are used in obtaining bounds on probabilities, in particular how they naturally appear in large deviation theory as rate functions. 
This course serves as a preparation for the course that will be given by F. Caravenna later at the school.

Marcel Ortgiese

Intermittency in branching random walks in random environment

We consider a branching random walk on the lattice, where the branching  rates are given by an i.i.d. (time-independent) potential. When keeping the potential fixed, the averaged number of particles satisfies the heat equation with a random potential known as the parabolic Anderson model. In the last twenty years, there has been

considerable progress in understanding the long-term behaviour of the averaged system. We concentrate on the situation when the potential is Pareto-distributed and we will discuss the effect of the averaging. In particular,  we will see that even though the qualitative picture is the same, the non-averaged system does not localize in the sites that are predicted

by the heat equation. 
(joint work with Matt Roberts)

Julien Poisat

Random pinning model with weakly correlated environment

The random pinning model is a model of Statistical Mechanics dealing with the (de)localization transition of a polymer interacting with an interface or with another polymer, via a disordered potential. The example of DNA denaturation is a motivation for studying the effect of correlations in the disorder sequence on the model. The purpose of this

talk is to give some results for weak correlations: in this regime, even if some quantities associated to the model are modified (such as critical point shifts), some qualitative features (such as the annealed critical exponent) are proved to remain unchanged. I will explain how one can

connect the decay of correlations with the regularity of some potential function, and how one can then use the spectral properties of an appropriate Ruelle-Perron-Frobenius operator to derive these results. 

Andrew Rechnitzer (mini course)

Enumerative combinatorics and models of polymers

Many problems in mathematics and physics - including many in the modeling of polymers - can be rephrased as "How many...".
Enumerative combinatorics seeks to answer these questions. Over the course of 2 or 3 lectures I will give a general introduction to the world of enumeration and the techniques of generating functions - including asymptotic methods championed by the late Philippe Flajolet.
In the second half of the short course I will show how we can apply these generating function methods to study polymers - especially adsorption, collapse and localisation.

Michele Salvi

Moment conditions for not-zero speed of random walks among random conductances 

Reversible random walks in random environment are called random walks among random conductances. It is well known that whenever one considers weights on the edges of the euclidean lattice (the conductances) bounded from above, the corresponding process has almost surely zero

limiting speed. We derive sharp log-moments conditions on the distribution of the conductances that force the random walk to have zero speed. We also show counterexamples -relying on geometrical constructions of random trees- of walks with positive speed, or even not-existing speed, whenever such conditions are not

fulfilled.
(joint work with Noam Berger, TU Munich)

Martin Slowik

Invariance principle for the random conductance model under moment conditions

We consider a continuous time random walk on the lattice $\mathbb{Z}^d$ in an environment of random conductances, $\mu_{x,y}$. The law of the environment is assumed to be ergodic with respect to space shifts with $\mathbb{P}[0 < \mu_{x,y} < \infty] = 1$.  In this talk, I will explain how a quenched invariance principle can be established under suitable moment conditions. A key ingredient in the proof is to establish the sub-linearity of the corrector by means of Moser's iteration scheme.
(joint work with Sebastian Andres (Univ. Bonn) and Jean-Dominique Deuschel (TU Berlin)).

Laurent Tournier

Quenched and annealed fluctuations of random walks in random environment, in connection to dynamics of polymer phase transitions

In this talk, we will present how some problems about random walks in one-dimensional random environment (RWRE) arose in the physics litterature in the context of phase transition in random polymers   (here, random polymers refer to chains of randomly chosen monomers,   like DNA,

rather than random spatial configurations of a chain), and   present recent results that provide a rigorous setting and new developments. We will be led to focus at the quenched distribution of the hitting times of an RWRE, as a quenched counterpart to a celebrated result by Kesten, Kozlov and Spitzer. This is a joint work with N. Enriquez, C. Sabot and O. Zindy.

Stu Whittington (mini course)

Polymer models and self-avoiding walks

The lectures will give an introduction to polymers and to some lattice models used to investigate their conformational properties. After a brief introduction to some classes of polymers and the kinds of questions that one would like to address, most of the time will be spent on 
self-avoiding walks. Some useful general techniques will be introduced in- cluding sub-additive inequalities, unfolding operations and pattern theorems. Applications will include models of polymer adsorption, polymers in confined geometries and random
knotting.

Tilman Wolff

Annealed asymptotics for occupation time measures of a random  walk among random conductances

The annealed asymptotic behaviour of local times, or occupation time meaasures, of a simple random walk is key to the long-time analysis of the solution to the parabolic Anderson problem, that is, the heat equation on the lattice with random potential.

Dependent on the tails of the potential distribution, it is crucial to establish large deviation principles for properly rescaled local times on fixed or time-dependent domains.
The random conductance model (RCM) describes a random walk with locally irregular diffusive dynamics, which seems in many cases more realistic than the homogeneous random walk. With a view to future analysis of the parabolic Anderson model with random conductances, it makes sense to

study the occupation time measures of a random walk among random conductances in terms of large deviation principles.  We work in the case of conductances that assume arbitrarily small values with exponentially small probability. Here, the scale of the corresponding large deviation principles is different from the SRW case.

We will focus on time-dependent domains and also address dimension-dependent properties of the correponding continuous variational problems.

PARTICIPANTS

PRACTICAL INFORMATION

Conference Location
The workshop location is Eurandom,  Den Dolech 2, 5612 AZ Eindhoven, METAFORUM, 4th floor, MF 12.

Eurandom is located on the campus of Eindhoven University of Technology, in the 'Laplacegebouw' building' (LG on the map). The university is located at 10 minutes walking distance from Eindhoven railway station (take the exit north side and walk towards the tall building on the right with the sign TU/e).

For all information on how to come to Eindhoven, please check http://www.eurandom.tue.nl/contact.htm

CONTACT
For more information please contact Mrs. Patty Koorn,
Coordinator of  EURANDOM

OTHER SPONSORS

Research Network Programme: RGLIS - Random Geometry of Large Interacting Systems and Statistical Physics

Last updated 01-07-13,
by PK