Correlation Decay for Maximum Weight Matchings on Sparse Graphs

TL;DR

This paper establishes exponential and polynomial decay of correlations for maximum weight matchings on sparse graphs with exponential edge weights, enabling analysis of infinite graphs and convergence properties.

Contribution

It proves correlation decay rates for maximum weight matchings on sparse graphs, extending understanding to infinite graphs and convergence behaviors.

Findings

Exponential decay of correlations on locally tree-like graphs.

Polynomial decay of correlations on graphs with degree at most three.

Existence and convergence results for maximum weight matchings on infinite graphs.

Abstract

We study correlation decay for the maximum weight matching problem on sparse graphs with i.i.d. edge weights. We show exponential decay of correlations when the underlying graphs are locally tree-like with uniformly bounded degree and the edge weights are exponential. We also prove a polynomial rate of decay of correlations for any finite graph with maximum degree at most three, again for exponential edge weights. As consequences of the correlation decay property, we obtain the existence of the maximum weight matching on infinite graphs, local weak convergence of the maximum weight matching, and a law of large numbers for its total weight.