An Invariance Principle for a Random Walk Among Moving Traps via Thermodynamic Formalism

Abstract

We consider a random walk among a Poisson cloud of moving traps on \(\mathbb {Z}^d\), where the walk is killed at a rate proportional to the number of traps occupying the same position. In dimension \(d=1\), we have previously shown that under the annealed law of the random walk conditioned on survival up to time t, the walk is sub-diffusive. Here we show that in \(d\geqslant 6\) and under diffusive scaling, this annealed law satisfies an invariance principle with a positive diffusion constant if the killing rate is small. Our proof is based on the theory of thermodynamic formalism, where we extend some classic results for Markov shifts with a finite alphabet and a potential of summable variation to the case of an uncountable non-compact alphabet.