Asymptotic Behavior of Branching Random Walks on Some Two-Dimensional Lattices
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ASYMPTOTIC BEHAVIOR OF BRANCHING RANDOM WALKS ON SOME TWO-DIMENSIONAL LATTICES K. S. Ryadovkin∗
UDC 519.21
A branching random walk on two-dimensional lattices corresponding to graphene and stanene is considered. It is assumed that sources of branching are located on lattices periodically. An asymptotics of the mean value of particles in each vertex of the lattice is obtained. Bibliography: 8 titles
1. Introduction It was shown in [3, 4] that spectral analysis of discrete periodic operators on the lattice Zd is important and useful tool for solving a probabilistic problem of studying the asymptotic behavior of a branching random walk with continuous time on Zd . Some results on the asymptotic behavior of a branching random walk with finite number of branching sources can be found in [3, 4] and the papers mentioned there. The general case of periodic random walk with branching was considered in [2]. In this paper, we found the leading term of the asymptotics as t → ∞ of the average number of particles M (v, u, t) at the vertex u at time t provided that at zero time there is exactly one particle at the vertex v. In the present paper, we use this general result to calculate the leading term of the asymptotic behavior of the average number of particles for specific graphs. The graphs considered in the paper correspond to the crystal lattices of graphene and stanene in the tight-binding model. In this model, the crystals are modeled by periodic discrete graphs, and the electromagnetic interaction is modeled by the discrete Schr¨ odinger operator. It should be noted that when considering a random walk on a graph, the transition intensity matrix of the random walk corresponds to the discrete Schr¨ odinger operator, and the potential determines the intensity of branching in the sources. In the second section we present the results of [2] without proof. The third and fourth sections are devoted to study the asymptotic behavior of the average number of particles for branching random walks on the graphene and stanene lattices, respectively.
2. Asymptotic behavior of the average number of particles We need the main result (Theorem 6) of paper [2]. It is formulated here not in full generality, but only for the case of walk on the lattice Z2 and finite number of possible transitions from each vertex. Let G = (Z2 , E) be a graph with vertex set Z2 and the set E of undirected edges. Assume that G is connected, has no loops, and is invariant under shifts by noncollinear vectors g1 , g2 ∈ Z2 (that is, the vectors g1 , g2 are periods of G). We also assume that the graph G is locally finite, that is, in any circle of finite radius only a finite number of edges can begin. We denote by κv the number of edges from E going out of the vertex v. ∗
St.Petersburg Department of the Steklov Mathematical Institute; St.Petersburg State University, St.Petersburg, Russia, e-mail: [email protected].
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 474, 2018, pp. 213–221. Original article submitted October 29, 2018. 1072-3374/20/2511-0141 ©20
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