Critical Site Percolation in High Dimension

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.

A branching process with deletions and mergers that matches the threshold for hypercube percolation

“…We begin by proving (5). For n ≥ 1 and u ∈ I n−1 , by our construction in Section 4.1 and by (16) we have…”

“…The study of the percolation threshold for Z d , and other infinite lattices, has a long history in both the mathematics and the physics literature, see e.g. [13,16] and the references therein. The study of percolation in the hypercube started with the analysis of the connectivity probability [11,3] and bounds on the threshold for the emergence of a linear-sized component [1,6].…”

Expansion for the critical point of site percolation: the first three terms

“…Theorem 1.1 heavily builds upon the results obtained in [13]. We use Section 2 to collect the necessary notation and results from [13] in order to prove our main result. At the heart of these results is an identity for τ p .…”

“…The lace expansion provides an expression for p c in terms of lace-expansion coefficients, which are defined in Definition 2.5. Moreover, it provides good control over these coefficients, and the results of [13] identify already the leading order term in (1.3).…”

“…The key technical tool for our approach is the lace expansion for site percolation. It was established in a recent paper [13], which itself draws its inspiration from Hara and Slade's seminal paper [11]. The lace expansion provides an expression for p c in terms of lace-expansion coefficients, which are defined in Definition 2.5.…”