Generalized approach to the non-backtracking lace expansion

TL;DR

This paper presents a generalized framework for the non-backtracking lace expansion, a technique used to analyze critical phenomena in spatial processes, providing conditions for its effectiveness across various models.

Contribution

It introduces a recursive formula for the non-backtracking lace expansion and states conditions under which it guarantees the infrared bound, broadening its applicability.

Findings

Conditions established for percolation in dimensions ≥11

Conditions established for lattice trees in dimensions ≥16

Conditions established for lattice animals in dimensions ≥21

Abstract

The lace expansion is a powerful perturbative technique to analyze the critical behavior of random spatial processes such as the self-avoiding walk, percolation and lattice trees and animals. The non-backtracking lace expansion (NoBLE) is a modification that allows us to improve its applicability in the nearest-neighbor setting on the $\mathbb Z^d$-lattice for percolation, lattice trees and lattice animals. The NoBLE gives rise to a recursive formula that we study in this paper at a general level. We state assumptions that guarantee that the solution of this recursive formula satisfies the infrared bound. In two related papers, we show that these conditions are satisfied for percolation in $d\geq 11$, for lattice trees in $d\geq 16$ and for lattice animals in $d\geq 21$.