A rate for the average vacancy of uniformly random matchings in linear hypergraphs

Abstract

Let $M$ be chosen uniformly from all matchings of a finite linear $k$-uniform hypergraph $H$, and let $\overline{q}(H)$ be the average probability that a vertex is left uncovered. If $H$ has maximum degree $D$ and normalized average degree $β=k|E(H)|/(|V(H)|D)$, then, for every fixed $k\geq 2$ and uniformly in the order, $\overline{q}(H) \leq 1-β+(βk+o_D(1))\log\log D/\log D$. The underlying estimate is the order-uniform count $\log Z(H)\geq (|E(H)|/D)(\log D-(k+o_D(1))\log\log D)$, and it also counts matchings of size $(1-o_D(1))|E(H)|/D$. We prove this by sampling edges, deleting those incident with unusually large sampled degrees, and controlling only the total deleted mass before applying the Molloy-Reed list edge-colouring theorem. For $d$-regular $H$, a quantitative version of the earlier Asratian-Kuzjurin sampling-to-counting route, using the near-perfect-matching theorem of Gould and Kelly, sharpens the coefficient: for every fixed $k\geq 3$, $\sup_H \overline{q}(H) \leq (\max\{3,k-1\}+o_d(1))\log\log d/\log d$. Kahn's stronger pointwise prediction was disproved by Lee for $k\geq 3$. The qualitative averaged conclusion was already recorded by Kahn and Kim, crediting Anders Johansson, and also follows from the Grable-Asratian-Kuzjurin enumeration; neither source states a rate.