Rapid mixing of the down-up walk on matchings of a fixed size

TL;DR

This paper proves that the down-up walk on matchings of fixed size k ≤ (1−δ)m^*(G) mixes in polynomial time on general graphs, advancing toward the optimal O_{Δ,δ}(n log n) bound conjectured for bounded-degree graphs. The authors achieve this via a novel multicommodity-flow construction that lower-bounds the spectral gap, avoiding the spectral-independence framework and instead giving explicit flow encodings with good and bad pair decompositions. They derive concrete mixing-time bounds: in general, τ_mix = O(n^{4/δ} m^4 k log(1/ε)), and for graphs with maximum degree Δ, τ_mix = O_{Δ,δ}(n^6 k log(1/ε)). The work also highlights fundamental obstacles for applying spectral-independence techniques to fixed-size matchings, using explicit examples to show that pinning can disrupt ergodicity and slack in the residual graph, thereby clarifying the limits of that approach.

Abstract

Let $G = (V,E)$ be a graph on $n$ vertices and let $m^*(G)$ denote the size of a maximum matching in $G$. We show that for any $δ> 0$ and for any $1 \leq k \leq (1-δ)m^*(G)$, the down-up walk on matchings of size $k$ in $G$ mixes in time polynomial in $n$. Previously, polynomial mixing was not known even for graphs with maximum degree $Δ$, and our result makes progress on a conjecture of Jain, Perkins, Sah, and Sawhney [STOC, 2022] that the down-up walk mixes in optimal time $O_{Δ,δ}(n\log{n})$. In contrast with recent works analyzing mixing of down-up walks in various settings using the spectral independence framework, we bound the spectral gap by constructing and analyzing a suitable multi-commodity flow. In fact, we present constructions demonstrating the limitations of the spectral independence approach in our setting.

Theorems & Definitions (10)

• Theorem 1.1
• Remark
• Definition 2.1
• Theorem 2.2
• Proposition 3.1
• proof
• Claim 3.2
• proof
• Claim 3.3
• proof