On the number of perfect matchings in random lifts

TL;DR

This paper investigates the distribution of the number of perfect matchings in random lifts of a fixed multigraph, providing asymptotic formulas and analyzing interactions with short cycles, using advanced probabilistic and combinatorial methods.

Contribution

It introduces new asymptotic formulas for the expectation of perfect matchings in random lifts of regular multigraphs and develops a theorem for summation estimation using Laplace's method.

Findings

Asymptotic expectation formula for perfect matchings in d-regular multigraphs

Analysis of interaction between perfect matchings and short cycles

Partial results on the second moment for specific multigraphs

Abstract

Let G be a fixed connected multigraph with no loops. A random n-lift of G is obtained by replacing each vertex of G by a set of n vertices (where these sets are pairwise disjoint) and replacing each edge by a randomly chosen perfect matching between the n-sets corresponding to the endpoints of the edge. Let X_G be the number of perfect matchings in a random lift of G. We study the distribution of X_G in the limit as n tends to infinity, using the small subgraph conditioning method. We present several results including an asymptotic formula for the expectation of X_G when G is d-regular, d\geq 3. The interaction of perfect matchings with short cycles in random lifts of regular multigraphs is also analysed. Partial calculations are performed for the second moment of X_G, with full details given for two example multigraphs, including the complete graph K_4. To assist in our calculations we provide a theorem for estimating a summation over multiple dimensions using Laplace's method. This result is phrased as a summation over lattice points, and may prove useful in future applications.