Continuity for the Rate Function of the Simple Random Walk on Supercritical Percolation Clusters
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Continuity for the Rate Function of the Simple Random Walk on Supercritical Percolation Clusters Naoki Kubota1 Received: 20 November 2018 / Revised: 20 November 2018 © Springer Science+Business Media, LLC, part of Springer Nature 2019
Abstract We consider the simple random walk on supercritical percolation clusters in the multidimensional cubic lattice. In this model, a quenched large deviation principle holds for the position of the random walk. Its rate function depends on the law of the percolation configuration, and the aim of this paper is to study the continuity of the rate function in the law. To do this, it is useful that the rate function is expressed by the so-called Lyapunov exponent, which is the asymptotic cost paid by the random walk for traveling in a landscape of percolation configurations. In this context, we first observe the continuity of the Lyapunov exponent in the law of the percolation configuration and then lift it to the rate function. Keywords Percolation · Random walk · Random environment · Large deviations · Lyapunov exponent Mathematics Subject Classification 60K37 · 60F10
1 Introduction 1.1 The Model For d ≥ 2, we denote by Zd the d-dimensional cubic lattice. Furthermore, E d is the set of all nearest-neighbor edges in Zd , i.e., E d := {x, y} ⊂ Zd : x − y1 = 1 ,
Naoki Kubota was supported by JSPS Grant-in-Aid for Young Scientists (B) 16K17620.
B 1
Naoki Kubota [email protected] College of Science and Technology, Nihon University, 24-1, Narashinodai 7-chome, Funabashi-shi, Chiba 274-8501, Japan
123
Journal of Theoretical Probability
where · 1 is the 1 -norm on Rd . Let ω = (ω(e))e∈E d denote a family of independent random variables satisfying P p (ω(e) = 1) = 1 − P p (ω(e) = 0) = p ∈ [0, 1]. An edge e ∈ E d is called open if ω(e) = 1, and closed otherwise. We say that a lattice path is open if it uses only open edges. Then, the chemical distance d(x, y) = dω (x, y) between x and y is defined by the minimal length of an open lattice path from x to y in the percolation configuration ω. For x ∈ Zd , we denote by Cx = Cx (ω) the open cluster containing x, i.e., the set of all vertices which are linked to x by an open lattice path. It is well known that there exists pc = pc (d) ∈ (0, 1) such that P p -almost surely, we have a unique infinite open cluster C∞ = C∞ (ω) with P p (0 ∈ C∞ ) > 0 whenever p ∈ ( pc , 1] (see Theorems 1.10 and 8.1 of [14] for instance). Let O = O(ω) be the set of all vertices which are endpoints of open edges. Define for x, y ∈ O with x − y1 = 1, πω (x, y) :=
1 1{ω({x,y})=1} ,1 . ∈ 2d {x,y }∈E d 1{ω({x,y })=1}
Then, the (discrete time) simple random walk on percolation clusters is the Markov z chain ((X n )∞ n=0 , (Pω )z∈O ) on O with the transition probabilities Pωz (X 0 = z) = 1, Pωz (X n+1 = y|X n = x) = πω (x, y). For each x ∈ Zd , denote by H (x) the first passage time through x, i.e., H (x) := inf{n ≥ 0 : X n = x}. Then, for any λ ≥ 0 and x, y ∈ O, we define the travel cost aλ (x, y) = aλ (x, y, ω) from x to y as aλ (
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