# European Institute for Statistics, Probability, Stochastic Operations Research and its Applications

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workshop

"Dynamic Random Environments"

Programme

 December 14 December 15 December 16 December 17 December 18 09.00-10.00 A.-S. Sznitman On the connectivity of the vacant set of random interlacements C. Liverani Random walks in evolving environments 10.00-10.15 Opening Break Break Break Break 10.15-11.15 B. Toth Brownian Random Polymers E. Bolthausen On the TAP equations and the high temperature regime in spin glasses D. Dawson Catalytic branching: reaction networks and spatial models 11.15-11.30 Break Break Break Break Break 11.30-12.30 A. Montanari T.b.a. P. Mörters Random networks with nonlinear preferential attachment 12.30-14.00 Lunch Lunch Lunch Lunch 14.00-15.00 A. Ramirez The generalized excited random walk on Zd 15.00-15.15 Break Break Break Break 15.15-16.15 16.15-16.30 Break Break Break 16.30-17.30 S. Friedli Scaling limit of the prudent walk G. Ben Arous Universality of superaging for spin glass dynamics 18.00 Workshop Dinner Restaurant “La Biba’

L. Avena

Random walks in dynamic random environments: LLN and LDP

We consider a class of one-dimensional interacting particle systems in equilibrium on the integer lattice, constituting a dynamic Random Environment (RE), together with a nearest-neighbor random walk that on occupied/vacant sites has a local drift to the right/left. We adapt a regeneration-time argument originally developed by Comets and Zeitouni for static RE to prove that, under a space-time mixing condition on the dynamic RE, the random walk has an a.s. asymptotic speed. Under extra assumptions on the RE, we prove a quenched and an annealed large deviation principle for the empirical speed of the walker. We discuss extensions of the previous results and open problems.

Presentation

G. Ben Arous
Joint Work with Onur Gun

Universality of superaging for spin glass dynamics

J. Berestycki
Joint work with N. Berestycki and J. Schweinsberg

The genealogy of branching Brownian motion with absorption

We consider a system of particles which perform branching Brownian motion with negative drift and are killed upon reaching zero, in the near-critical regime where the total population stays roughly constant with approximately $N$ particles. We show that the characteristic time scale for the evolution of this population is of order $(\log N)^3$, in the sense that when time is measured in these units, the scaled number of particles converges to a variant of Neveu's continuous-state branching process. Furthermore, the genealogy of the particles is then governed by a coalescent process known as the Bolthausen-Sznitman coalescent. This validates the non-rigorous predictions by Brunet, Derrida, Muller, and Munier for a closely related model.

Presentation

E. Bolthausen

On the TAP equations and the high temperature regime in spin glasses

Presentation

A. Bovier
Joint work with Nicolas Champagnat and Anne Pohrt

Survival of the misfit: Life in times of change

I discuss an individual based model for adaptive dynamics of biological populations under the influence of death, reproduction, mutation and competition. I explain how mutations can be crucial for global long-term survival in time-dependent environments. This will be illustrated in a simple two state example.

Presentation

J. Cerny
Joint work with A. Teixeira and D. Windisch

Phase transition for the complement of random walk trajectory on random regular graphs

We consider a simple random walk on the random regular graph.  We show that the complement of the trajectory of the walk stopped after a time proportional to the size of the graph exhibits a phase transition similar to Bernoulli percolation.

Presentation

D. Dawson

Catalytic branching: reaction networks and spatial models

Presentation

M. Deijfen

Invariant random graphs with prescribed iid degrees

Models for generating random graphs with prescribed degree distribution have been extensively studied the last few years. Most existing models for this purpose however do not take spatial aspects into account, that is, there is no metric defined on the vertex set. I will discuss spatial versions of the problem. More precisely, given a degree distribution F and a spatial vertex set - for instance Z^d or the points of a spatial Poisson process - how should one go about to obtain a translation invariant random graph on the given vertex set with degree distribution F? Which properties do the resulting configurations have? I will describe existing results and a number of open problems.

R. Fontes

On the dynamics of trap models in $\Z^d$

We consider symmetric trap models in $\Z^d$ and derive a scaling limit for the depth of the currently visited trap at large times. This is given by the size of the jump of a stable subordinator seen at the inverse of an associated stable subordinator. This is a self similar process of index 1, which explains the aging behavior of those processes. Conditions on the embedded random walk include all transient ones, all planar ones, and also some one dimensional recurrent ones. The argument is based on a special construction of the environment on the range of the random walks, so it holds in distribution with respect to the original variables. Under a further condition, this is strengthened to a weak version of convergence in probability.

Presentation

S. Friedli
Joint work with V. Beffara and Y. Velenik

Scaling limit of the prudent walk

We describe the scaling limit of the nearest neighbour prudent walk on the square lattice, which performs steps uniformly in direc- tions in which it does not see sites already visited. We show that the process eventually settles in one of the quadrants, and derive its scaling limit, which can be expressed in terms of a pair of indepen- dent stable subordinators. We also show that the asymptotic speed of the walk is well-defined in the $L^1$-norm and equals $3/7$.

B. Gentz
Joint work with Florent Barret, Nils Berglund and Bastien Fernandez

Metastability in a chain of bistable systems

We will discuss metastability in a system of interacting diffusions. While for weak coupling, this system behaves like an Ising spin system with Glauber dynamics, it tends to a Ginzburg--Landau SPDE as coupling strength and particle number go to infinity in a suitable way. The transition between these regimes involves a sequence of symmetry-breaking bifurcations.

The classical Wentzell--Freidlin theory allows to describe the optimal transition paths and yields the exponential asymptotics of transition times between metastable states. Away from bifurcations, work by Bovier, Gayrard, Eckhoff and Klein on the Eyring--Kramers formula yields the subexponential asymptotics. We will show how the subexponential asymptotics of these transition times changes in the presence of non-quadratic saddles as encountered at critical coupling strengths.

Finally we will address the question of metastable transition times in the limiting Ginzburg--Landau SPDE.

Presentation

G. Kozma
Joint work with Asaf Nachmias

Arm exponents for high dimensional percolation

We examine critical percolation when the dimension is large enough, or when the dimension is > 6 and the lattice is sufficiently spread out. We show that the probability of an open path between the center of a ball of radius r and its boundary is ~ r^(-2). The proof uses results achieved using lace expansion (specifically the determination of the two-point function) but does not use lace expansion directly.

Presentation

A. Le Ny

Transient random walk in dimension 2

We present a family of randomly oriented versions of the square lattice Z^2, whose horizontal orientations can be generated by dynamical systems, on which the simple random walk appears to be transient and to satisfy a non-standard functional limit theorem with an unconventional normalization in n^3/4, whose limiting object is related to a non-Gaussian self-similar process already present in the context of random walk in random sceneries introduced by Kesten and Spitzer in 1979.

C. Liverani
Joint work with D. Dolgopyat and G. Keller

Random walks in evolving environments

I will describe some results on random walks in mixing environments. As my motivations in the subject are slightly unconventional, I will also spend some words illustrating them. I will conclude with some open problems.

G. Maillard
Joint work with J. Gärtner and F. den Hollander

The parabolic Anderson model in voter environment

We consider the parabolic Anderson model $\partial_t u = \kappa\Delta u + \xi u$ on $\mathbb{Z}^d\times [0,\infty)$ with a space-time dependent homogeneous ergodic random field $\xi$. To study intermittency in terms of the moments of the solution $u$, we investigate the behavior of the annealed Lyapunov exponents as a function of the dimension and the diffusion constant $\kappa$. In this talk we focus on the case where $\xi$ is the voter model with opinions $0$ and $1$ that are updated according to a random walk transition kernel, starting from either the Bernoulli or the equilibrium measure. One of the main obstacles is the non-reversibility of the voter dynamics, since it precludes the application of spectral techniques. The duality with coalescing random walks is key to our analysis, and leads to a representation formula for the Lyapunov exponents that allows for the application of large deviation estimates.

A. Montanari

T.b.a.

P. Mörters
Joint work with Steffen Dereich (Marburg).

Random networks with nonlinear preferential attachment

We define a dynamic model of random networks, where new vertices are connected to old ones with a probability proportional to a nonlinear function of their degree. Our main interest is in the phase transitions occuring when we vary the attachment function and move from weak to strong preferential attachment. Properties discussed include the degree distribution, existence of a hub and clustering.

Presentation

C. Newman
Joint work with K. Ravishankar and E. Schertzer.

The Brownian web and the scaling limit of the one-dimensional stochastic Potts model

The Brownian web (BW) is a version of the continuum scaling limit of the family of one dimensional simple symmetric random walks starting from all points of (discrete) space-time, originally analyzed by R. Arratia and by B. Toth and W. Werner. In a recent paper (arXiv:0806.0158, to appear in Annales de l'IHP), we studied Poissonian markings of (1,2) points of the BW and their relation to the Brownian net (BN) of R. Sun and J. Swart. (1,2) points are the nongeneric points in (continuous) space-time where one path enters from earlier time and two paths leave, only one of which is a continuation of the earlier path.

In this talk, we will explain how marked (1,2) (and (0,2)) points can be used to construct the scaling limit of a stochastic one-dimensional q-state Potts model. Earlier work (joint with L.R. Fontes, M. Isopi and K. Ravishankar in Annales de l'IHP 42 (2006) 37-60) concerned scaling limits of the stochastic Ising model where q=2 and the (1,2) points play no role because there is no possibility for a third "color" to be nucleated at the boundary between two color regions.

A. Ramirez

The generalized excited random walk on Zd

Presentation

F. Redig
Joint work with L. Avena and F. den Hollander

Random walks in dynamic random environments: a perturbative approach

We consider a random walk driven by a dynamics in a strongly ergodic regime. We show that if the coupling between the walk and the environment is small, then there exists a unique ergodic stationary measure for the environment process, and expectations of local functions in this measure can be expressed as a power series in the coupling. As a consequence, we derive a LLN, a CLT and large deviation properties.

Presentation

B. Toth
Joint work with R. Tarres, and B. Valko, respectively, with I. Horvath, B. Veto

Brownian Random Polymers

I will give a survey of recent results about the asymptotics of the Brownian Random Polymer process introduced by Durrett and Rogers in 1992. The process is pushed by some signed average of its own occupation time measure. In the most interesting self-repelling cases it is driven by a smeared out negative gradient of its local time. The process in the continuous space-time analogue of the so-called "myopic (or true) self-avoiding random walk".

I will present two classes of results:

(1) In 1d I will show various diffusive and superdiffusive bounds, depending on the infrared asymptotics of the driving function.

(2) In three and more dimensions we prove a diffusive limit (CLT) for the self-repelling case. On the way to the CLT we relax Varadhan's sector condition.

T. Seppalainen

Scaling exponents for a one-dimensional directed polymer

We study a 1+1-dimensional directed polymer in a random environment on the integer lattice with log-gamma distributed weights and both endpoints of the polymer path fixed. We show that under appropriate boundary conditions the fluctuation exponents for the free energy and the polymer path take the values conjectured in the theoretical physics literature.

Presentation

R. Sun
Joint work in progress with Jan  M. Swart (UTIA, Prague) and Emmanuel Schertzer (Columbia, New York)

Stochastic flows of kernels in the Brownian net and the Brownian web

We identify a suitable scaling limit of one-dimensional random walks in i.i.d. space-time random environments as a stochastic flow of kernels in the Brownian net and the Brownian web. The notion of stochastic flows of kernels was introduced by Le Jan and Raimond, where they showed that each consistent family of n-point motions gives rise to a stochastic flow of kernels, which can be loosely interpreted as the transition kernels of a random motion in a random environment with independent innovations. They constructed a specific class of flows on R, which was recently generalized by Howitt and Warren to a much larger class we call the Howitt-Warren flows, where the n-point motions are sticky Brownian motions that arise as the scaling limit of the n-point motions for random walks in i.i.d. space- time random environments. Here we give a graphical construction of the underlying environment for the Howitt-Warren flow in terms of the Brownian net (resp. the Brownian web), which loosely speaking consists of a collection of branching-coalescing (resp. coalescing) Brownian motions starting from every point in the space-time plane. Almost sure path properties for the Howitt-Warren flow are derived based on the graphical construction.

A.-S. Sznitman

On the connectivity of the vacant set of random interlacements

E. Vares

A system of grabbing particles related to Galton-Watson trees

In this lecture I will discuss a particle system that can be seen as a simple toy model for polymerization. The lecture is based on a joint article with Jean Bertoin and Vladas Sidoravicius (to appear in Random Structures and Algorithms).

We start with a finite system of particles with arms that are activated randomly to grab other particles. The following two rules are imposed:

once a particle has been grabbed then it cannot be grabbed again, and an arm cannot grab a particle that belongs to its own cluster. We are interested in the shape of a typical polymer in the situation when the initial number of monomers is large and the numbers of arms of monomers are given by i.i.d.

random variables. Our main result is a limit theorem for the empirical distribution of polymers, where limit is expressed in terms of a Galton-Watson tree.

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