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.

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

“…It will be clear that sufficient control over the coefficients will result in the expansion of Theorem 1.1. In fact, the results from [13] immediately give the first term of (1.3).…”

“…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 .…”

“…We use this section to state the definitions and results from [13] needed in the proof of Theorem 1.1.…”

“…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.…”