# Fall School on Statistical Mechanics and 5th Annual PhD Student Conference in Probability

## Abstracts for the talks

**Christian Bartsch** *Universität Münster* (joint work with *Michael Kochler, Thomas Kochler*)**Survival of a Branching Random Walk in Random Environment**

We consider a particular Branching Random Walk in Random Environment (BRWRE)

on started with one particle at the origin. Particles reproduce

according to an offspring distribution (which depends on the location) and

move either one step to the right (with a probability in (0,1] which may

also depend on the location) or stay in the same place. We give criteria for

local and global survival and show that global survival is equivalent to

exponential growth of the moments.

**Marek Biskup** *UCLA***The Random Conductance Model**

In the random conductance model, each edge of the hypercubic

lattice is assigned a positive and finite random variable called

the conductance. The resulting (random) resistor network is directly

linked to a Markov chain --- referred to, invariably, as a random walk

among random conductances --- where at each step the walk chooses

a neighbor at random with probability proportional to the conductance

of the corresponding edge. The conductances are distributed according

to a translation-invariant, or even iid, law; the properties of the

random

walk are studied against a typical sample from this law (these are the

so called quenched problems).

In my three lectures I will discuss the following facts/situations:

(1) The proof of recurrence and transience for this random walk and

the connection with effective resistivity.

(2) The proof of an invariance principle for the path of the random

walk in the case when the conductances are bounded away from

zero and infinity (so called elliptic case).

(3) Extensions to non-elliptic situations (random walk on the

supercritical

percolation cluster, regular and anomalous heat-kernel decay).

I will finish with an outlook of problems awaiting solutions and

possible

directions of future research. My contributions to this field are

based on

joint works with N. Berger, T. Prescott, G. Kozma and C. Hoffman.

**Alex Drewitz** *TU Berlin***Ballisticity Criteria for Random Walk in Random Environments**

Consider a random walk in a uniformly elliptic i.i.d.

random environment in dimensions . In

2002, Sznitman introduced for each the ballisticity

conditions and (*T*'), the latter being defined as the

fulfilment of

for all . He proved that (*T*') implies ballisticity and

that for each , is equivalent

to (*T*'). It is conjectured that this equivalence holds for all

. Here we prove that for

, where is a dimension dependent

constant

taking values in the interval (0.366,0.388), is

equivalent to (*T*').

**Felix Rubin** *University of Zurich***Scaled limit for the largest eigenvalue from the generalized Cauchy random matrix ensemble**

In this talk, we are interested in the asymptotic properties for

the largest

eigenvalue of the Hermitian random matrix ensemble,

called the Generalized Cauchy ensemble *GCy*, whose eigenvalues PDF is

given by

where *s* is a complex number such that and where *N* is the

size of

the matrix ensemble. We will see that for this

ensemble, the appropriately rescaled largest eigenvalue converges in law.

We also

express the limiting probability distribution in

terms of some

non-linear second order differential equation. Eventually, we show that the

convergence of the probability distribution function of

the re-scaled largest eigenvalue to the limiting one is at least of order

(1/*N*).

**Herbert Spohn** *TU München***Stochastic Growth Processes**

In my minicourse I explain growth models in the Kardar-Parisi-Zhang

universality class

in one spatial dimension. In particular I cover various growth models,

their stationary measures,

scaling behavior, the link to tiling problems, directed last passage

percolation, and to random matrices.

**Philipp Thomann** *Universität Zürich***On a Generalization of the Sherrington-Kirkpatrick model**

Many calculations on the SK-Model use its quite special properties. In

order to understand the problem more thoroughly we treat a generalized

version where spins are not constrained to Ising type. We calculate

its free energy by use of the ideas appearing in Talagrand (2009).

## Date and Location

**September 07 - 12, 2009**

Max Planck Institute for Mathematics in the Sciences

Inselstraße 22

04103 Leipzig

Germany

see travel instructions

## Scientific Organizers

**Mathias Becker**

Universität Leipzig

**Wolfgang König**

Universität Leipzig

**Chiranjib Mukherjee**

Max-Planck-Institut für Mathematik in den Naturwissenschaften

## Administrative Contact

**Katja Bieling**

Max Planck Institute for Mathematics in the Sciences