Critical Site Percolation in High Dimension
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Critical Site Percolation in High Dimension Markus Heydenreich1
· Kilian Matzke1
Received: 15 November 2019 / Accepted: 29 June 2020 © The Author(s) 2020
Abstract We use the lace expansion to prove an infra-red bound for site percolation on the hypercubic lattice in high dimension. This implies the triangle condition and allows us to derive several critical exponents that characterize mean-field behavior in high dimensions. Keywords Site percolation · Lace expansion · Mean-field behavior · Infra-red bound Mathematics Subject Classification 60K35 · 82B43
1 Introduction 1.1 Site Percolation on the Hypercubic Lattice We consider site percolation on the hypercubic lattice Zd , where sites are independently occupied with probability p ∈ [0, 1], and otherwise vacant. More formally, for p ∈ [0, 1], Zd we consider the probability space (, F , P p ), where = {0, 1} , the σ -algebra F is generated by the cylinder sets, and P p = x∈Zd Ber( p) is a product-Bernoulli measure. We call ω ∈ a configuration and say that a site x ∈ Zd is occupied in ω if ω(x) = 1. If ω(x) = 0, we say that the site x is vacant. For convenience, we identify ω with the set of occupied sites {x ∈ Zd : ω(x) = 1}. Given a configuration ω, we say that two points x = y ∈ Zd are connected and write x ←→ y if there is an occupied path between x and y—that is, there are points d x = v0 , . . . , vk = y in Zd with k ∈ N0 := N∪{0} such that |vi −vi−1 | = 1 (with |y| = i=1 |yi | the 1-norm) for all 1 ≤ i ≤ k, and vi ∈ ω for 1 ≤ i ≤ k − 1 (i.e., all inner sites are occupied). Two neighbors are automatically connected (i.e., {x ←→ y} = for all x, y with |x − y| = 1). Many authors prefer a different definition of connectivity by requiring
Communicated by Hal Tasaki.
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Markus Heydenreich [email protected] Kilian Matzke [email protected]
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Mathematisches Institut, Ludwig-Maximilians-Universität München, Theresienstr. 39, 80333 Munich, Germany
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M. Heydenreich, K. Matzke
both endpoints to be occupied as well. These two notions are closely related and we explain our choice in Sect. 1.4. Moreover, we adopt the convention that {x ←→ x} = ∅, that is, x is not connected to itself. We define the cluster of x to be C (x) := {x} ∪ {y ∈ ω : x ←→ y}. Note that apart from x itself, points in C (x) need to be occupied. We also define the expected cluster size (or susceptibility) χ( p) = E p [|C (0)|], where for a set A ⊆ Zd , we let |A| denote the cardinality of A, and 0 the origin in Zd . We define the two-point function τ p : Zd → [0, 1] by τ p (x) := P p (0 ←→ x). The percolation probability is defined as θ ( p) := P p (0 ←→ ∞) = P p (|C (0)| = ∞). We note that p → θ ( p) is increasing and define the critical point for θ as pc = pc (Zd ) = inf{ p > 0 : θ ( p) > 0}. Note that we can define a critical point pc (G) for any graph G. As we only concern ourselves with Zd , we write pc or pc (d) the refer to the critical point of Zd .
1.2 Main Result The triangle condition is a versatile criterion for several critical exponents to exist and to take on th
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