*Survival probability of a test particle in a random vicious
environment. (Joint work with S. Krishnamurthy, R. Rajesh and R. Tribe.)*

**Abstract:** We study the survival probability of a test
particle placed in the system of interacting random walkers on
a $d$-dimensional
lattice. It is assumed that random walkers diffuse at a rate $D$ and
the test particle diffuses at a rate $\delta D$. Two random walkers
at the same lattice site can either annihilate each other or coagulate
to form a single walker at rates $\lambda_{a}, \lambda_{c}$
correspondingly. The test
particle is consumed
by a random walker positioned at the same lattice site at a rate
$\lambda$. We compute the large time asymptotics of the
survival probability of
the test particle as a function of time $t$ and the parameters
introduced above.

We approach the problem by establishing a rigorous correspondence between the Markov chain describing our reaction-diffusion system and a system of SPDE's with imaginary mutiplicative noise (Lee-Cardy equations.

We then study the large time asymptotics of solutions to these equations using the (non-rigorous) perturbative renormalization group method.

In the talk we will present a detailed comparison of our answers with exact results and/or with results of numerical simulations. We will also discuss the application of our work to the study of various persistence phenomena in the $Q$-state Potts model.