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Correlation Decay for Maximum Weight Matchings on Sparse Graphs

Wai-Kit Lam, Arnab Sen

TL;DR

This work analyzes correlation decay for the maximum weight matching on sparse graphs with i.i.d. edge weights, establishing precise decay bounds that depend on graph structure and using a contraction argument built around the exponential weight memoryless property. The authors derive a cavity-based recursion for the edge-acceptance probability and a log-transform to obtain contractivity, proving exponential decay on locally tree-like graphs and a polynomial decay rate for max-degree-3 graphs. Consequences include the existence of a well-defined MWM on infinite graphs, stability under local weak convergence, and a law of large numbers for the total MWM weight; these results offer rigorous insight into replica-symmetric behavior of MWM in sparse regimes and enable limiting results for random graphs. The methods combine deterministic MWM structure (bonuses, augmenting paths) with probabilistic contraction arguments, yielding explicit bounds and broad applicability to infinite-volume and local-convergence settings.

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.

Correlation Decay for Maximum Weight Matchings on Sparse Graphs

TL;DR

This work analyzes correlation decay for the maximum weight matching on sparse graphs with i.i.d. edge weights, establishing precise decay bounds that depend on graph structure and using a contraction argument built around the exponential weight memoryless property. The authors derive a cavity-based recursion for the edge-acceptance probability and a log-transform to obtain contractivity, proving exponential decay on locally tree-like graphs and a polynomial decay rate for max-degree-3 graphs. Consequences include the existence of a well-defined MWM on infinite graphs, stability under local weak convergence, and a law of large numbers for the total MWM weight; these results offer rigorous insight into replica-symmetric behavior of MWM in sparse regimes and enable limiting results for random graphs. The methods combine deterministic MWM structure (bonuses, augmenting paths) with probabilistic contraction arguments, yielding explicit bounds and broad applicability to infinite-volume and local-convergence settings.

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.

Paper Structure

This paper contains 18 sections, 11 theorems, 137 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. A brief sketch of proof
  4. Organization of the paper
  5. Preliminaries
  6. Restriction of the maximum weight matching
  7. Augmenting paths
  8. Cavity recursion of MWM
  9. Characterizing edges in MWM
  10. Local bounds for bonuses
  11. Correlation decay
  12. Correlation decay on a tree
  13. Proof of Theorem \ref{['thm: temp_corr_decay']}
  14. Proof of Theorem \ref{['thm: corr_decay_degree_3']}
  15. Maximum weight matching on an infinite graph
  16. ...and 3 more sections

Key Result

Theorem 1.3

Let $G$ be a connected graph whose degree is bounded above by $D$. Suppose that the edge weights are i.i.d. exponential random variables with mean $1$. Let $e \in E(G)$ and $r\geq 3$ be such that $\mathbb{B}_e^r(G)$ is a tree. Then we have

Figures (2)

  • Figure 1: In (a), the blue subgraph is $G\mathbin{\fgebackslash} \{u\}$. In this case, its connected component containing $v$ overlaps with $G\mathbin{\fgebackslash} \{(uv)\}$, so the bonuses $\mathsf{B}(u, G \mathbin{\fgebackslash} \{(uv)\})$ and $\mathsf{B}(v, G\mathbin{\fgebackslash}\{u\})$ are not independent of each other, but they are independent of the edge weight $w_{(uv)}$. In (b), the graph $G$ is a tree, and in this case $\mathsf{B}(u, G \mathbin{\fgebackslash} \{(uv)\})$ is exactly $\mathsf{B}(u, T_{v\to u})$; similarly, $\mathsf{B}(v, G\mathbin{\fgebackslash}\{u\}) = \mathsf{B}(v, T_{u\to v})$.
  • Figure 2: Illustration of the recursion \ref{['eq:rec_p']} on the edges of $T$

Theorems & Definitions (20)

  • Theorem 1.3
  • Theorem 1.4
  • Proposition 1.6: MWM on an infinite graph
  • Proposition 1.7: Local weak convergence of MWM
  • Corollary 1.8
  • proof
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Definition 2.4
  • ...and 10 more