Correlation bound for distant parts of factor of IID processes

TL;DR

This paper establishes a quantitative bound on the correlation decay between distant parts of factor of i.i.d. processes on regular trees, showing they are nearly uncorrelated as the distance increases.

Contribution

It provides a new explicit bound on the correlation between distant subgraphs in factor of i.i.d. processes on regular trees, extending understanding of their tail triviality.

Findings

Correlation between distant parts decays exponentially with distance

Correlation bound is proportional to k(d-1)/(\sqrt{d-1})^k

Supports the triviality of 1-ended tails in such processes

Abstract

We study factor of i.i.d. processes on the $d$-regular tree for $d \geq 3$. We show that if such a process is restricted to two distant connected subgraphs of the tree, then the two parts are basically uncorrelated. More precisely, any functions of the two parts have correlation at most $k(d-1) / (\sqrt{d-1})^k$, where $k$ denotes the distance of the subgraphs. This result can be considered as a quantitative version of the fact that factor of i.i.d. processes have trivial 1-ended tails.