On large girth regular graphs and random processes on trees

TL;DR

This paper investigates invariant random processes on regular trees, introduces a new class called typical processes, and uses these concepts to derive combinatorial results about random regular graphs.

Contribution

It introduces the class of typical processes and establishes a correspondence principle linking random regular graphs with ergodic theory on trees.

Findings

Provided a sufficient condition for branching Markov chains to be factor of i.i.d.

Developed a new framework connecting random graphs and ergodic theory.

Proved combinatorial statements about random regular graphs using entropy inequalities.

Abstract

We study various classes of random processes defined on the regular tree $T_d$ that are invariant under the automorphism group of $T_d$. Most important ones are factor of i.i.d. processes (randomized local algorithms), branching Markov chains and a new class that we call typical processes. Using Glauber dynamics on processes we give a sufficient condition for a branching Markov chain to be factor of i.i.d. Typical processes are defined in a way that they create a correspondence principle between random $d$-reguar graphs and ergodic theory on $T_d$. Using this correspondence principle together with entropy inequalities for typical processes we prove a family of combinatorial statements about random $d$-regular graphs.