On infinite-volume mixing

TL;DR

This paper explores the concept of mixing in infinite ergodic theory, proposing a new approach for observables with infinite-volume averages, and tests these ideas on infinite measure-preserving systems like random walks.

Contribution

It introduces a novel notion of mixing for infinite-volume observables and evaluates different mathematical definitions within the context of infinite measure-preserving systems.

Findings

Proposes a new framework for mixing in infinite ergodic theory.

Analyzes the suitability of various definitions of mixing for infinite systems.

Demonstrates the approach on random walk models.

Abstract

In the context of the long-standing issue of mixing in infinite ergodic theory, we introduce the idea of mixing for observables possessing an infinite-volume average. The idea is borrowed from statistical mechanics and appears to be relevant, at least for extended systems with a direct physical interpretation. We discuss the pros and cons of a few mathematical definitions that can be devised, testing them on a prototypical class of infinite measure-preserving dynamical systems, namely, the random walks.