Unlacing the lace expansion: a survey to hypercube percolation

TL;DR

This paper surveys the study of percolation phase transition on the hypercube, highlighting alternative proof techniques to the lace-expansion method and discussing their potential applications to high-dimensional tori.

Contribution

It provides a comprehensive survey of hypercube percolation and introduces a new simple proof for the triangle condition without lace-expansion, offering tools for future research.

Findings

Survey of hypercube percolation studies

New simple proof of the triangle condition

Potential applications to high-dimensional tori

Abstract

The purpose of this note is twofold. First, we survey the study of the percolation phase transition on the Hamming hypercube {0,1}^m obtained in the series of papers [9,10,11,24]. Secondly, we explain how this study can be performed without the use of the so-called "lace-expansion" technique. To that aim, we provide a novel simple proof that the triangle condition holds at the critical probability. We hope that some of these techniques will be useful to obtain non-perturbative proofs in the analogous, yet much more difficult study on high-dimensional tori.