Ergodicity Criterion for One-Sided, One-Dimensional IPS with a Long-Lived State

TL;DR

This paper establishes an ergodicity criterion for one-sided, one-dimensional interacting particle systems based on the existence of a long-lived state, providing new insights into the dynamics and stability of such systems.

Contribution

It introduces a novel ergodicity criterion for 1D IPS with positive rates, based on the ratio of exit to entry rates, extending understanding of the Positive Rates Conjecture.

Findings

Ergodicity occurs when the ratio of exit to entry rates is below √2.

The criterion applies to a broad class of IPS, including models related to the East model.

The approach uses canonical coupling to identify large agreeing blocks in spacetime.

Abstract

We study one-dimensional, one-sided, nearest-neighbor Interacting Particle Systems (IPS) with positive rates and identify a criterion for ergodicity based on the presence of a long lived state a site can occupy. The criterion is that the IPS admits a state with a ratio of rates of exiting to entering below~$\sqrt{2}$. The main idea is that under the canonical coupling, it is possible to identify large blocks in spacetime where two coupled trajectories must agree to share the special state, which inhibits the spread of disagreement. The result covers much of the parameter region left open by earlier work on the Positive Rates Conjecture (PRC) for simple IPS, narrowing the unresolved IPS to noisy versions of the East model.