# Entropy dissipation estimates for inhomogeneous zero-range processes

## TL;DR

This paper establishes a dimension-free modified log-Sobolev inequality for inhomogeneous zero-range processes on complete graphs by leveraging coupling techniques, addressing a longstanding open problem in the field.

## Contribution

It introduces a novel coupling-based method to derive entropy dissipation estimates for heterogeneous zero-range processes, extending previous homogeneous results.

## Key findings

- Proves a dimension-free modified log-Sobolev inequality for inhomogeneous zero-range processes.
- Addresses an open problem from Caputo and Posta (2007).
- Method is simple and potentially widely applicable.

## Abstract

Introduced by Lu and Yau (CMP, 1993), the martingale decomposition method is a powerful recursive strategy that has produced sharp log-Sobolev inequalities for homogeneous particle systems. However, the intractability of certain covariance terms has so far precluded applications to heterogeneous models. Here we demonstrate that the existence of an appropriate coupling can be exploited to bypass this limitation effortlessly. Our main result is a dimension-free modified log-Sobolev inequality for zero-range processes on the complete graph, under the only requirement that all rate increments lie in a compact subset of $(0,\infty)$. This settles an open problem raised by Caputo and Posta (PTRF, 2007) and reiterated by Caputo, Dai Pra and Posta (AIHP, 2009). We believe that our approach is simple enough to be applicable to many systems.