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# YEP XII “Random walk in random environment”

## Mar 23, 2015, 08:00 - Aug 27, 2015, 17:00

#### Summary

Random walk in random environment (RWRE) has originally been introduced in the early second half of the last century as a model for motions in disordered media, for the replication of DNA chains, and for phase transitions in alloys, among others. While the area has initially developed not too fast, in particular the last two decades have resulted in significant progress and a much deeper understanding of the topic. In fact, it has attracted significant mathematical attention and has undergone a major development, where, among others, results on limiting velocities, scaling limits, and large deviations have been established.

The YEP XII brings together many junior and some senior participants whose research interests are strongly related to RWRE. This mix of researchers aims at giving a special, open atmosphere which in the past has proven to be very fruitful for fostering discussions among the participants.

#### Sponsors

#### Organisers

Alexander Drewitz | Columbia University |

Markus Heydenreich | LMU Munich |

#### Speakers

Mini courses

Noam Berger | The Hebrew University / TU Munich | |

David Croydon | University of Warwick | |

Vladas Sidoravicius | IMPA, Rio de Janeiro |

Invited speakers

Stein Bethuelsen | University of Leiden |

Oriane Blondel | CNRS, Université Lyon 1 |

Jiří Černý | University of Vienna |

Dirk Erhard | University of Warwick |

Onur Gün | WIAS Berlin |

Brett Kolesnik | UBC Vancouver |

Júlia Komjáthy | TU Eindhoven |

Sebastian Müller | LAPT, Marseilles |

Jan Nagel | TU Munich |

Tal Orenshtein | University Lyon 1 |

Alejandro Ramirez | Pontificia Universidad Catolica de Chile |

Pierre-Francois Rodriguez | ETH Zurich |

Ron Rosenthal | ETH Zurich |

Michele Salvi | TU Munich |

Martin Slowik | TU Berlin |

Francois Siemenhaus | Dauphine Université Paris |

Renato Soares dos Santos | WIAS Berlin |

Tobias Wassmer | University of Vienna |

Attila Yilmaz | Bogaziçi University |

#### Abstracts

**Random Walk on Attractive Spin-Flip Dynamics
**Consider random walk on the d-dimensional lattice whose transition probabilities are given by an underlying supercritical contact process. This is a primal example of random walk in dynamic random environment, which does not fall into the well-studied cone-mixing class. We derive a law of large number and derive bounds on large deviation probabilities for this model.

We obtain these results in broad generality for any attractive two-state dynamics when started from a trivial measure.

(joint work with Markus Heydenreich (Leiden))

**The strongly ballistic phase for random walk in random environment
**The general case of random walk in random environment (RWRE) in dimensions larger than one is not well understood. However, if one makes some extra assumptions on the environment, then one can use several tools to get a clearer understanding of the behavior of the random walk. There are several different sets of assumptions, each enabling the use of a different set of tools, under which we can obtain interesting results. The purpose of this course is to focus on one such set of assumptions, known as strong ballisticity, and describe the progress in recent years in the understanding of the process under this set of assumptions.

**Random walks on the East model
**The East model is a one-dimensional interacting particle system with non attractive spin-flip dynamics. In the physics literature, it is a key example of a model with glassy features. Here we take this model as a random environment and investigate the behaviour of two different random walks whose jump rates depend on the current configuration. In particular, we are interested in the relation between the dynamics of the East model and the asymptotic speed and diffusion coefficient of the random walks.

(part of the talk will be based on a joint work with Luca Avena and Alessandra Faggionato)

This contribution received funding from the INTEGER project

**Ancestral lineages in spatial populations: Over the oriented random walk on oriented percolation cluster
**We study the long time behaviour of ancestral lineages in spatial population models. These can be viewed as a random walk in a particular type of Markovian random environment. The question is whether these lineages satisfy the CLT.

(joint work, partly in progress, with M. Birkner, N. Gantert and Andrej Depperschmidt)

**Scaling limits of random walks on critical random trees and graphs
**This course will survey some recent work regarding the scaling limits of random walks on various models of random trees and other graphs at criticality, including critical Galton-Watson trees, the critical Erdos-Renyi random graph and the uniform spanning tree in two dimensions. In each of these examples, the random structure considered exhibits some kind of interesting fractal behaviour in the limit. This leads to topological challenges in showing the associated random walks converge to a diffusion, and also gives rise to anomalous features for both the discrete and continuous processes (such as sub-Gaussian transition density estimates).

**The parabolic Anderson model in a dynamic random environment: random conductances
**The parabolic Anderson model is a differential equation, which describes the evolution of a field of particles performing independent nearest neighbor simple random walks with binary branching: particles jump at rate $\kappa$, $\kappa > 0$, split into two and die at rates determined by the environment. We denote by u(x,t) the mean number of particles at site x at time t conditioned on the evolution of the environment. In this talk I will discuss a modified version of this model. More precisely I introduce the case in which the random walks move according to random walks among a field of random conductances ${\kappa(x,y)}_{x,y\in \Z^d}$. For specific choices of the random environment I will discuss a link between the exponential growth rate of the number of particles at the origin for the original model to the one of the modified model.

This is joint work in progress with Frank den Hollander (Leiden) and Gregory Maillard (Marseille).

**Branching random walks in random environments on hypercubes
**We study branching random walks on hypercubes with spatially random branching rates. We explicitly describe the asymptotics of the expected number of particles, which corresponds to mutation-selection models on random fitness landscapes, for various random environments, in particular, for Gaussian landscapes as in the Random Energy Model (REM) of spin glasses.

(joint work with Luca Avena and Marion Hesse)

**The Cut Locus of the Brownian Map: Continuity and Stability
**The Brownian map is a random geodesic metric space which is homeomorphic to the sphere, of Hausdorff dimension 4, and the scaling limit of a wide variety of planar maps. In a sense, it is a purely random, non-differentiable surface. We strengthen the so-called confluence of geodesics phenomenon observed at the root of the Brownian map, and with this, reveal several properties of its rich geodesic structure.

Our main result is the continuity of the cut locus on an open, dense subset of the Brownian map. Moreover, the cut locus is uniformly stable, in the sense that any two cut loci coincide outside a nowhere dense set. (joint work with Omer Angel (UBC) and Grégory Miermont (ENS Lyon & IUF))

**Fixed speed competition on the configuration model with infinite variance degrees
**In this talk we consider competition of two spreading colors starting from single sources on the configuration model with i.i.d.\ degrees following a power-law distribution with exponent \( \tau\in (2,3) \).

In this model two colors spread with a fixed but not necessarily equal speed on the unweighted random graph. We answer the question how many vertices the two colors paint eventually. When the speeds are not equal, then the faster color paints almost all vertices. When the speeds are equal, we show that coexistence sensitively depends on the initial local neighbourhoods of the source vertices. This reinforces the common sense that speed and location are very important features in advertising.

**Rotor-routing on Galton-Watson trees
**A rotor-router walk on a graph is a deterministic process, in which each vertex is endowed with a rotor that points to one of the neighbors. A particle located at some vertex first rotates the rotor in a prescribed order, and then is routed to the neighbor the rotor is now pointing at. In the talk we discuss the behavior of rotor-router walks on Galton-Watson trees and give a classification in recurrence and transience for transfinite rotor-router walks on these trees.

(joint work with W. Huss and E. Sava-Huss)

**The Einstein relation in the random conductance model
**The Einstein relation says that for a motion under external force the derivative of the effective velocity at 0 (as a function of the strength of the force) is given by the diffusivity of the unperturbed motion.

While successfully applied in both theoretical and experimental physics, rigorous proofs exist only for few models. We prove the Einstein relation for the random walk in Z^d, when the transition probabilities are determined by random conductances of the edges, chosen independent and identically distributed and bounded away from 0 and infinity.

The proof is based on Lebowitz and Rost’s argument using Girsanov’s theorem to obtain an alternative description of the diffusivity and a regeneration structure for the biased process, which is robust for small values of the bias.

(joint work with Nina Gantert and Xiaoqin Guo)

**Excited random walk with periodic cookies
**We will discuss excited random walk on the integers in elliptic and identically piled environments with periodic cookies.

This is a discrete time process on the integers defined by parameters $p_1,…,p_M$ in $(0,1)$ for some positive integer $M$, where in the $i$-th visit to an integer $z$ the walker moves to $z+1$ with probability $p_{i \mod M}$, and to $z−1$ with probability $1-p_{i \mod M}$. The main result will be presented is an explicit formula, in terms of $p_1,…,p_M$, for determining recurrence, transience to the left, or transience to the right. As an application one can easily construct transient walks even when the average drift per period is zero.

(joint work with Gady Kozma and Igor Shinkar)

**Quenched central limit theorem for random walk in ergodic space-time environment
**We prove a quenched central limit theorem for random walk in a time dependent random environment under a mild ergodicity assumption.

(joint work with Xiaoqin Guo and Jean-Dominique Deuschel)

**On level-set percolation for the Gaussian free field
**We investigate the percolation model obtained by considering level sets of the Gaussian free field on the d-dimensional lattice above a given height h. We will discuss some of the recent progress in the study of its phase transition. An application of our results to a specific random conductance model will also be mentioned.

(partly based on joint work with A.-S. Sznitman and A. Drewitz)

**Quenched invariance principle for simple random walk on clusters of correlated percolation models
**We derive a quenched invariance principle for simple random walk on the unique infinite cluster for a general class of percolation models on $\mathbb{Z}^d$, $d\geq2$.

This includes models with long-range correlations such as random interlacements in dimension $d\geq3$ at every level, as well as for the vacant set of random

interlacements and the level sets of the Gaussian free field in the regime of the so-called local uniqueness.

An essential ingredient of our proof is a new isoperimetric inequality for this type of correlated percolation models.

(joint work with Eviatar Procaccia and Artëm Sapozhnikov)

**The law of large numbers for the Variable Range Hopping model
**The Variable Range Hopping model is considered in Physics as an accurate representation of electrical conduction in semiconductors. From the mathematical point of view, it represents a prominent example of reversible long-range random walks on random point processes, which generalize in several ways the classical random conductance model on the lattice. We ask ourselves how an external field influences the limiting velocity of the walk: So far, only very few models of biased random walks with trapping mechanisms have been rigorously studied. A precise control of the speed is also a fundamental step towards the proof of an Einstein Relation for this model.

This is an ongoing project together with Alessandra Faggionato and Nina Gantert

**Random walks in dynamic environment – mutual interaction case
**During three lectures I will discuss a couple of examples of RWDRE, where environment is evolving in time and is affected by the walker. Few methods, including multi scale arguments and geometric Markov chain approach will be presented, as well as many open problems.

**Random walk driven by simple exclusion process
**We prove strong law of large numbers and an annealed invariance principle for a random walk in a one-dimensional dynamic random environment evolving as the simple exclusion process with jump parameter $\gamma$.

First we establish that, if the asymptotic velocity of the walker is non-zero in the limiting case “$\gamma = \infty$” where the environment gets fully refreshed between each step, then, for $\gamma$ large enough, the walker still has a non-zero asymptotic velocity in the same direction.

Second we establish that if the walker is transient in the limiting case $\gamma = 0$, then, for $\gamma$ small enough but positive, the walker has a non-zero asymptotic velocity in the direction of the transience.

These two limiting velocities can sometimes be of opposite sign.

In all cases, we show that fluctuations are normal.

**Random conductance model in a degenerate ergodic environment: Invariance principle and heat kernel behaviour
**Consider a continuous time random walk on the Euclidean lattice $\mathbb{Z}^d$ in an environment of random conductances taking values in $[0, \infty)$. The law of the environment is assumed to be ergodic with respect to space shifts and satisfies some moment conditions. In this talk, I will discuss recent results on a quenched invariance principle and local limit theorem for this Markov process. In particular, I will explain how a quenched invariance principle can be established in case the law of the random conductances has an atom at zero.

(joint work with Sebastian Andres (Univ. Bonn), Jean-Dominique Deuschel (TU Berlin) and Tuan Ahn Nguyen (TU Berlin))

**Mass concentration in the parabolic Anderson model with doubly-exponential tails
**We consider the solution of the heat equation with random potential on the d-dimensional lattice with initial condition localised at the origin. The potential is supposed i.i.d. with upper tails close to doubly-exponential.

In this case, the solution is known to exhibit intermittent behaviour, i.e., its mass is asymptotically concentrated on relatively small “islands” that are well-separated in space. The number of islands needed is known to be a.s. asymptotically bounded by any small power of time. We show that, with probability tending to one as time increases, most of the mass of the solution is carried by a single island of bounded size. A crucial ingredient in the proof is the recent characterization due to Biskup and Koenig of the max-order class of the principal eigenvalue of the Anderson Hamiltonian in a growing box.

(joint work with Marek Biskup and Wolfgang Koenig)

**Aging of the Metropolis dynamics on the Random Energy Model
**Aging is one of the interesting features appearing in the long-time behavior of complex disordered systems such as spin glasses. The dynamics of these systems can be described by certain Markov chains in random environments. In the last two decades many works have proved aging for different spin glass models. These works however mostly lack some realism, as the dynamics considered are just a time change of simple random walk. In my talk I will present our recent progress in proving aging for the physically more realistic Metropolis dynamics. Joint work with Jiří Černý.

**Variational formulas and disorder regimes of random walks in random potentials
**I will start by providing three variational formulas for the quenched free energy of a random walk in random potential (RWRP) when the underlying walk is directed or undirected, the environment is stationary & ergodic, and the potential is allowed to depend on the next step of the walk which covers RWRE. Next, in the directed i.i.d. case, I will give two variational formulas for the annealed free energy of RWRP. These five formulas are the same except that they involve infima over different sets, and I will say a few words about how they are derived. Then, I will present connections between the existence & uniqueness of the minimizers of these variational formulas and the weak & strong disorder regimes of RWRP. I will end with a conjecture regarding very strong disorder.

(joint work with Firas Rassoul-Agha and Timo Seppalainen)