On the Number of Perfect Matchings in Random Lifts

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 ≥ 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.

The Bethe Permanent of a Nonnegative Matrix

“…For this, we use results from [16] that give a combinatorial characterization of the Bethe partition function of a factor graph in terms of the partition function of graph covers of this factor graph. Interestingly, very similar objects were considered by Greenhill et al [35]; we will comment on this connection in Section VII-E.…”

“…Interestingly enough, as shown by the authors of [35], for any , one can give a high-order approximation of and, therefore, of the degree-Bethe partition function [16] . For the corresponding expressions, we refer the interested reader to [35].…”

“…After the initial submission of this paper, we became aware of the paper by Greenhill et al [35] on counting perfect matchings in random graph covers. Using the findings of [16] and this paper, their results can, once they have been translated to factor graphs, be seen as defining an NFG with and computing , along with approximately computing .…”

“…8(a). In terms of factor graphs, the paper [35] considers the NFG shown in Fig. 8(b): like it has function nodes on the left-hand side and function nodes on the right-hand side.…”

“…1) Translating the results of a paper by Greenhill et al [35] to graphical models, it turns out that the authors compute a high-order approximation to the quantity for some NFG with . The NFG is in general different from , where the latter NFG was specified in Definition 4.…”