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11:15-13:00:
Vladas Sidoravicius
On stochastic growth
in Random Environment
14:15-16:00: Franz
Merkl
Recent results on edge reinforced random walks, (Joint
work with Silke Rolles, UCLA)
Abstracts:
Vladas Sidoravicius
On stochastic growth in
Random Environment
Consider the family of stochastic growth processes,
which can be described as follows:
There is a ``gas'' of $A$-particles, each of which performs a continuous time simple
random walk on $\Bbb Z^d$, with jumprate $D_A$. In addition, initially there are finitely many
$B$-particles which perform continuous time simple random walks with jumprate $D_B$. The only interaction is that when a
$B$-particle and an $A$-particle coincide (or become close enough to each other),
the latter instantaneously turns into a $B$-particle. All $B$-particles recuperate
(that is, turn back into $A$-particles) independently of each other at a rate $\lambda$.
This family is extremely reach and, depending on the values of $D_A$ and $D_B$, one obtains broad spectrum of models, such as DLA-type systems, or systems which are known
in the literature as ``stochastic Manna sandpiles'' and belong to the class of so-called Self Organized
Criticality models.
Most of these systems lack good subadditive properties. I will discuss new robust
multiscale renormalization methods which permit to treat some of them. In particular, I
will illustrate in case $D_A=D_B>0$ (and no recuperation), how to obtain some basic estimates for
the growth of the set $\widetilde B(t):= \{x \in \Bbb Z^d:$ a $B$-particle visits
$x$ during $[0,t]$\}, and then we will show that $B(t) = \widetilde B(t) + [-\frac 12, \frac 12]^d$ grows linearly in time with
an asymptotic shape, i.e., there exists a non-random set $B_0$ such that $(1/t)B(t)
\to B_0$, in a sense which will be made precise.
Next we show that there is a critical recuperation rate $\lambda_c > 0$ such that the $B$-particles survive (globally) with positive probability if $\lambda < \lambda_c$ and die out with probability 1 if $\lambda > \lambda_c$.
Franz
Merkl
Recent results on edge
reinforced random walks, (Joint work
with Silke Rolles, UCLA)
Edge reinforced random walks were
introduced in the eighties by
Persi Diaconis. In the talk, I will describe recent results
about linearly edge reinforced random walks on
infinite ladders:
Convergence to a stationary distribution, and exponential bounds for
the distribution of the
location of the the random walker. The proofs are based on an
representation of the reinforced random walk on an infinite ladder as
a random walk in random environment. The random environment has a
complicated dependence structure; it is given by a marginal of an infinite-volume
Gibbs measure. This Gibbs measure is analyzed using entropy
estimates and deformation arguments from equilibrium
statistical mechanics.