# 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