Ramanujan graphings and correlation decay in local algorithms

TL;DR

This paper establishes optimal bounds on correlation decay in local algorithms on large-girth regular graphs and relates these bounds to Ramanujan graphings, providing insights into their spectral properties.

Contribution

It introduces a new correlation decay bound for local algorithms on large-girth regular graphs and characterizes the correlation sequences for factor of i.i.d processes on the infinite regular tree.

Findings

Proves an optimal upper bound for correlation decay in large-girth regular graphs.

Shows that the correlation bounds are tight and achieved by certain processes.

Provides an explicit description of all possible correlation sequences for factor of i.i.d processes.

Abstract

Let $G$ be a large-girth $d$-regular graph and $\mu$ be a random process on the vertices of $G$ produced by a randomized local algorithm. We prove the upper bound $(k+1-2k/d)\Bigl(\frac{1}{\sqrt{d-1}}\Bigr)^k$ for the (absolute value of the) correlation of values on pairs of vertices of distance $k$ and show that this bound is optimal. The same results hold automatically for factor of i.i.d processes on the $d$-regular tree. In that case we give an explicit description for the (closure) of all possible correlation sequences. Our proof is based on the fact that the Bernoulli graphing of the infinite $d$-regular tree has spectral radius $2\sqrt{d-1}$. Graphings with this spectral gap are infinite analogues of finite Ramanujan graphs and they are interesting on their own right.