Typicality and entropy of processes on infinite trees

TL;DR

This paper explores the entropy of invariant processes on infinite trees, linking typical properties of large random regular graphs to entropy measures, and extends results to Galton-Watson trees.

Contribution

It introduces a micro-state entropy concept for invariant processes on infinite trees and provides new conditions for typicality, extending to Galton-Watson trees.

Findings

Entropy inequalities yield new sufficient conditions for typicality.

Micro-state entropy refines the understanding of typical processes.

Results connect entropy with asymptotic free energy in statistical physics.

Abstract

Consider a uniformly sampled random $d$-regular graph on $n$ vertices. If $d$ is fixed and $n$ goes to $\infty$ then we can relate typical (large probability) properties of such random graph to a family of invariant random processes (called "typical" processes) on the infinite $d$-regular tree $T_d$. This correspondence between ergodic theory on $T_d$ and random regular graphs is already proven to be fruitful in both directions. This paper continues the investigation of typical processes with a special emphasis on entropy. We study a natural notion of micro-state entropy for invariant processes on $T_d$. It serves as a quantitative refinement of the notion of typicality and is tightly connected to the asymptotic free energy in statistical physics. Using entropy inequalities, we provide new sufficient conditions for typicality for edge Markov processes. We also extend these notions and results to processes on unimodular Galton-Watson random trees.