An Invariance Principle for a Random Walk Among Moving Traps via Thermodynamic Formalism
Siva Athreya, Alexander Drewitz, Rongfeng Sun
TL;DR
The paper analyzes a simple random walk on ${\mathbb Z}^d$ interacting with a Poisson cloud of mobile traps, where the walker is killed at a rate $\gamma$ times the local trap count. By recasting the trapping problem in thermodynamic formalism on an uncountable, non-compact alphabet and proving a Ruelle–Perron–Frobenius-type result with summable variation, the authors construct an $h$-transformed ergodic Markov process and derive a mixing bound in $L^2$. This framework yields a functional central limit theorem for the annealed path conditioned on survival in dimensions $d\ge 6$, with a diffusion coefficient $\sigma^2>0$ for sufficiently small $\gamma$. In contrast to the subdiffusive behavior known in one dimension, the results show diffusive fluctuations in high dimensions and highlight the utility of extending thermodynamic-formalism to uncountable alphabets; the analysis also provides tools that may apply to other non-compact settings in statistical mechanics and probability.
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 $d\geq 6$ 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.