Dominating occupancy processes by the independent site approximation

TL;DR

This paper compares occupancy processes with independent site models, establishing conditions where the former is stochastically dominated by the latter, with implications for spin systems.

Contribution

It introduces a framework to compare occupancy processes to independent site models and proves dominance conditions, extending to spin systems.

Findings

Occupancy processes can be stochastically dominated by independent site models under certain conditions.

The results apply to a broad class of Markov chains on binary state spaces.

Extensions to spin systems are achieved via limiting arguments.

Abstract

Occupancy processes are a broad class of discrete time Markov chains on $\{0,1\}^{n}$ encompassing models from diverse areas. This model is compared to a collection of $n$ independent Markov chains on $\{0,1\}$, which we call the independent site model. We establish conditions under which an occupancy process is smaller in the lower orthant order than the independent site model. An analogous result for spin systems follows by a limiting argument.}