Annihilating random walks and perfect matchings of planar graphs

Annihilating random walks (ARW) have been studied in the early 70’s within the theory of interacting particles systems (see [1],[2],[3] and [9]). The idea was to study a system of particles moving on a graph according to certain laws of attraction. The system we study in this paper is defined as follows: the initial system consists of particles at every site of 2 . Then, each particle simultaneously performs discrete simple random walk on , that is, every particle, independently of each other, has probability one half of taking its next step to the left and one half to the right. If two particles are at the same time on the same position they annihilate each other. Let x and σ x 1 if there is a particle on site x and σ x 0 if not. Then, for all T , the positions of particles at time T is described by σT 0 1 Ω. ARW is then a discrete time Markov chain with state space Ω. Many results concerning ARW can be found in the literature but most of them for continuous time systems. We give here some results for ARW on with discrete time and this is obtained by a one-to-one correspondence with a statistical mechanics model: the dimer model on planar graphs. This model was first studied in the physical literature by P.W. Kasteleyn, H.T emperley and M. Fisher in the 60’s ([4],[5]) and then in the mathematical community by R. Kenyon ([6],[7]) in the late 90’s and we should note here (see [8]) that this statistical mechanical model has already been useful to understand other discrete random walks, among them loop erased random walks. These links enable one to understand random walks with enumerative combinatorial tools and gives an algebraic aspect to the theory. Given a graph G the dimer model studies the set M G of perfect matchings of G, that is the set of families of edges of G such that every vertex of G lies in exactly one edge of the family (these edges are called dimers). We will describe a graph G such that every configuration of trajectories of ARW correspond to exactly one perfect matching of G. Understanding the global and local statistics of M G leads then to results on ARW.