Atoms of the matching measure

  title={Atoms of the matching measure},
  author={Ferenc Bencs and Andr'as M'esz'aros},
  journal={Electronic Journal of Probability},
We prove that the matching measure of an infinite vertex-transitive connected graph has no atoms. Generalizing the results of Salez, we show that for an ergodic non-amenable unimodular random rooted graph with uniformly bounded degrees, the matching measure has only finitely many atoms. Ku and Chen proved the analogue of the Gallai-Edmonds structure theorem for non-zero roots of the matching polynomial for finite graphs. We extend their results for infinite graphs. We also show that the… 

Tables from this paper

A refined Gallai-Edmonds structure theorem for weighted matching polynomials



Invariant random matchings in Cayley graphs

We prove that any non-amenable Cayley graph admits a factor of IID perfect matching. We also show that any connected d-regular vertex tran- sitive graph admits a perfect matching. The two results

Mean quantum percolation

We study the spectrum of adjacency matrices of random graphs. We develop two techniques to lower bound the mass of the continuous part of the spectral measure or the density of states. As an

An analogue of the Gallai-Edmonds Structure Theorem for non-zero roots of the matching polynomial

Matchings in Benjamini-Schramm convergent graph sequences

The analytic approach yields a short proof for the Nguyen-Onak (also Elek--Lippner) theorem saying that the matching ratio is estimable, and proves the slightly stronger result that the independence ratio are estimable for claw-free graphs.

On Quantum Percolation in Finite Regular Graphs

The aim of this paper is twofold. First, we study eigenvalues and eigenvectors of the adjacency matrix of a bond percolation graph when the base graph is finite and well approximated locally by an

Interlacing Families I: Bipartite Ramanujan Graphs of All Degrees

The existence of infinite families of (c, d)-biregular bipartite graphs with all non-trivial eigenvalues bounded by √c-1+√d-1, for all c, d ≥ q 3 is proved.

Matchings on infinite graphs

Elek and Lippner (Proc. Am. Math. Soc. 138(8), 2939–2947, 2010) showed that the convergence of a sequence of bounded-degree graphs implies the existence of a limit for the proportion of vertices

Matching Measure, Benjamini–Schramm Convergence and the Monomer–Dimer Free Energy

The matching measure of a lattice L is defined as the spectral measure of the tree of self-avoiding walks in L, and an analytic advantage of the matching measure over the Mayer series leads to new, rigorous bounds on the monomer–dimer free energies of various Euclidean lattices.

Every totally real algebraic integer is a tree eigenvalue

  • J. Salez
  • Mathematics
    J. Comb. Theory, Ser. B
  • 2015

Paths, Trees, and Flowers

  • J. Edmonds
  • Mathematics
    Canadian Journal of Mathematics
  • 1965
A graph G for purposes here is a finite set of elements called vertices and a finite set of elements called edges such that each edge meets exactly two vertices, called the end-points of the edge. An