A note on the dimensional crossover critical exponent
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A note on the dimensional crossover critical exponent Pablo A. Gomes1
· Rémy Sanchis1
· Roger W. C. Silva2
Received: 19 December 2019 / Revised: 26 September 2020 / Accepted: 16 October 2020 © Springer Nature B.V. 2020
Abstract We consider independent anisotropic bond percolation on Zd ×Zs where edges parallel to Zd are open with probability p < pc (Zd ) and edges parallel to Zs are open with probability q, independently of all others. We prove that percolation occurs for q ≥ 8d 2 ( pc (Zd ) − p). This fact implies that the so-called Dimensional Crossover critical exponent, if it exists, is greater or equal than 1. In particular, using known results, we conclude the proof that, for d ≥ 11, the crossover critical exponent exists and equals 1. Keywords Dimensional crossover · Anisotropic percolation · Critical threshold · Phase diagram Mathematics Subject Classification 60K35 · 82B43 · 82B26
1 Introduction and results 1.1 Background In this note, we consider anisotropic bond percolation on the graph (Zd+s , E(Zd+s )), where E(Zd+s ) is the set of edges between nearest neighbors of Zd+s . We simplify notation and denote this graph by Zd+s = Zd × Zs . An edge of Zd+s is called a Zd -edge (respectively a Zs -edge) if it joins two vertices which differ only in their Zd (respectively Zs ) component. Formally, a Zd -edge is of the form (u d , u s ), (u d , u s ),
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Roger W. C. Silva [email protected] Pablo A. Gomes [email protected] Rémy Sanchis [email protected]
1
Departamento de Matemática, Universidade Federal de Minas Gerais, Belo Horizonte, Brazil
2
Departamento de Estatística, Universidade Federal de Minas Gerais, Belo Horizonte, Brazil
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where u d and u s should be understood as the components of u ∈ Zd+s . Probability is introduced as follows: given two parameters p, q ∈ [0, 1], we declare each Zd -edge open with probability p and each Zs -edge open with probability q, independently of all others. This model is described by the probability space (, F, P p,q ) where d+s = {0, 1} E(Z ) , F is the σ -algebra generated by the cylinder sets in and P p,q = e∈E μ(e), where μ(e) is Bernoulli measure with parameter p or q according to e been a Zd - edge or a Zs - edge, respectively. Given two vertices u, v ∈ Zd+s , we say that u and v are connected in the configuration ω if there exists an open path in Zd+s starting in u and ending in v. The event where v and u are connected is denoted by {ω ∈ : v ↔ u in ω}, and we write C(ω) = {u ∈ Zd+s : u ↔ 0 in ω} for the open cluster containing the origin. We denote by θ ( p, q) = P p,q (ω ∈ : |C(ω)| = ∞) the main macroscopic function in percolation theory and denote the mean size of the open cluster by χ ( p, q) = E p,q (|C(ω))|). Whenever necessary we shall write χ p (d) and pc (d) for the expected cluster size and critical threshold on Zd with a single parameter p ∈ (0, 1). For a comprehensive background in percolation theory, we refer the reader to [9]. It is easy to see, by a standard coupling argument, that θ ( p, q) is a monotone
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