Session:Random Walk in Random Environment <br>Organizer:

Saturday, August 6, 2005 - 1:30pm - 3:15pm
EE/CS 3-180
Ofer Zeitouni (University of Minnesota, Twin Cities)
  • Random walk in random scenery
    Nina Gantert (Universität Fridericiana (TH) Karlsruhe)
    Let $(Z_n)_{n\in N_0}$ be a $d$-dimensional random walk in random scenery, i.e., $Z_n=\sum_{k=0}^{n-1}Y(S_k)$ with $(S_k)_{k\in N_0}$ a random walk in $Z^d$ and $(Y(z))_{z\in Z^d}$ an i.i.d. scenery, independent of the walk. The walker's steps have mean zero and finite variance.

    We identify the speed and the rate of the logarithmic decay of $P(\frac 1n Z_n>b_n)$ for various choices of sequences $(b_n)_n$ in $[1,\infty)$. Depending on $(b_n)_n$ and the upper tails of the scenery, we identify different regimes for the speed of decay and different variational formulas for the rate functions.

    In contrast to recent work by A.~Asselah and F.~Castell, we consider sceneries unbounded to infinity. It turns out that there are interesting connections to large deviation properties of self-intersections of the walk, which have been studied recently by X. Chen. The talk is based on joint work with Wolfgang Koenig, Remco van der Hofstad and Zhan Shi.
  • Growth in dynamic random environment
    Vladas Sidoravicius (Institute of Pure and Applied Mathematics (IMPA))
    We consider the following model for the spread of an infection: There is a gas of so-called A-particles, each of which performs a continuous time simple random walk on Z^d, with jumprate D_A. We assume that initially the number of A-particles at x is independent, mean mu_A Poisson distribution. In addition, there are B-particles which perform continuous time simple random walks with jumprate D_B. Initially we start with only one B-particle in the system, located at the origin. The B-particles move independently of each other, and the only interaction is that when a B-particle and an A-particle coincide, the latter instantaneously turns into a B-particle. For different values of the parameters D_A and D_B one obtains several interesting evolutions, raging from stochastic sandpile type dynamics (activated random walkers) to contact process like evolution and DLA-type growth. The difficult aspect of all these models is the absence of useful subadditive quantities. I briefly discuss these models before going to our main results: Consider the set C(t):= {x in Z^d: a B-particle visits x during [0,t]}. If D_A=D_B, then B(t) := C(t) + [- 1/2, 1/2]^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. Moreover, if the recuperation transition form A to B occurs at the rate lambda>0, for each particle independently, then we show that there is a phase transition between survival and extiction of B particles. I also briefly present the key ideas of the proof.

    The talk is based on joint works with H. Kesten.
  • On some self-interacting random walks in random environment
    Martin Zerner (Eberhard-Karls-Universität Tübingen)