Matchings on infinite graphs

Abstract

Elek and Lippner (2010) showed that the convergence of a sequence of bounded-degree graphs implies the existence of a limit for the proportion of vertices covered by a maximum matching. We provide a characterization of the limiting parameter via a local recursion defined directly on the limit of the graph sequence. Interestingly, the recursion may admit multiple solutions, implying non-trivial long-range dependencies between the covered vertices. We overcome this lack of correlation decay by introducing a perturbative parameter (temperature), which we let progressively go to zero. This allows us to uniquely identify the correct solution. In the important case where the graph limit is a unimodular Galton-Watson tree, the recursion simplifies into a distributional equation that can be solved explicitly, leading to a new asymptotic formula that considerably extends the well-known one by Karp and Sipser for Erdős-Rényi random graphs.

Solution of the Monomer–Dimer Model on Locally Tree-Like Graphs. Rigorous Results

“…Analytic continuation techniques are used in order to extend results from "large" x to all positive x. Our results extend those of Bordenave, Lelarge, Salez [10] which are valid for graphs with bounded degree, and are generalised to arbitrary degree only for x → 0 (this is called the maximum matching problem and it is not treated in this paper, see instead [11] for its first solution in the Erdős-Rényi case and [12,13] for other generalisations). A complete theoretical physics picture of the monomer-dimer model (matching problem) on sparse random graphs was given by Zdeborová and Mézard in [14], where several quantities were computed including the pressure of the model, using the so called replica-symmetric version of the cavity method.…”

“…This lemma already appeared in [10] and in particular point ii can be seen also as a consequence of theorem 4.2 in [2].…”

“…Following [10] we introduce a recursion relation for the probability R x (·) that will be extensively used in the sequel; this is a rewriting of the recursion relation for the partition function Z · (x) present in [2].…”

“…In [10] it is shown how to rewrite the identity (3) in terms of the probability R x (G, o) of having a monomer in o. We use this form to deduce the distributional identity (1) for Y (x) := lim r→∞ R x (T (r), o), where T (r) is a random tree with root o, r generations and i.i.d.…”

The densest subgraph problem in sparse random graphs

“…This quickly leads to a proof of Theorem 1 (Section 7). In spirit, the role of the perturbative parameter ε > 0 is comparable to that of the temperature in [10], although no Gibbs-Boltzmann measure is involved in the present work.…”