Exact time dependence of the cumulants of a tracer position in a dense lattice gas

TL;DR

This paper introduces a general method to precisely compute the time evolution of cumulants for a tracer particle in dense lattice gases, covering various models and regimes, including short-time dynamics and biased movements.

Contribution

It presents a novel, exact analytical approach to determine the time dependence of tracer cumulants in dense lattice gases, extending beyond symmetric random walks to more complex scenarios.

Findings

Exact cumulant generating function derived for arbitrary times.

Method applicable to biased tracers, comb structures, and higher dimensions.

Provides insights into short-time tracer dynamics in dense environments.

Abstract

We develop a general method to calculate the exact time dependence of the cumulants of the position of a tracer particle in a dense lattice gas of hardcore particles. More precisely, we calculate the cumulant generating function associated with the position of a tagged particle at arbitrary time, and at leading order in the density of vacancies on the lattice. In particular, our approach gives access to the short-time dynamics of the cumulants of the tracer position -- a regime in which few results are known. The generality of our approach is demonstrated by showing that it goes beyond the case of a symmetric 1D random walk, and covers the important situations of (i) a biased tracer; (ii) comb-like structures; and (iii) $d$-dimensional situations.