October 8, 2002, 16.00-18.00 h.

EURANDOM Public lecture by

Professor Harry Kesten, EURANDOM chair

Particle systems with a collection of moving catalysts
(Joint work with Vladas Sidoravicius)

We consider two interacting particle systems which at first sight appear rather dissimilar. However, both reduce to the study of fluctuations in the density (in space-time) of a system of independent random walkers.

The simpler model can be regarded as a model for the spread of a rumor. The random walkers will be denoted as A-particles. They perform independent continuous time simple random walks on the d-dimensional integer lattice. In addition we start with one B-particle at the origin. This B-particle ``knows a rumor'' and performs also a random walk on the d-dimensional integer latice. Originally, the A-particles are ignorant of the rumor, but when an A-particle meets a B-particle it turns into a B-particle and also starts spreading the rumor. The principal question of interest is to describe how far the rumor has spread at a large time t.

The other model has the same A-system as in the rumor model, but now we start with infinitely many B-particles, say one at each site of the d-dimensional integer lattice. In addition to performing random walks, the B-particles can die (at a fixed rate) and split into two particles at a rate which is proportional to the number of A-particles at the same space-time position. Thus the B-particles form branching random walks in an environment which changes over space and time. The A-particles form a system of moving catalyst for the B-particles. In this model one wants to know for which values of the parameters the system of B-particles survives, respectively dies out.