On the number of perfect matchings in random lifts

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.