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

TL;DR

This paper introduces a graph process inspired by hypercube and lattice percolation, analyzing survival/extinction conditions that align with known critical probabilities, without providing a traditional branching process proof.

Contribution

It defines a new graph process with deletions and mergers that models hypercube and lattice percolation thresholds, offering insights into their critical behaviors.

Findings

Survival and extinction conditions match known percolation thresholds

Analysis accounts for dependencies between individuals in the process

Open questions remain about monotonicity of survival probability

Abstract

We define a graph process $\mathcal{G}(p,q)$ based on a discrete branching process with deletions and mergers, which is inspired by the 4-cycle structure of both the hypercube $Q_d$ and the lattice $\mathbb{Z}^d$ for large $d$. Individuals have Poisson offspring distribution with mean $1+p$ and certain deletions and mergers occur with probability $q$; these parameters correspond to the mean number of edges discovered from a given vertex in an exploration of a percolation cluster and to the probability that a non-backtracking path of length four closes a cycle, respectively. We prove survival and extinction under certain conditions on $p$ and $q$ that heuristically match the known expansions of the critical probabilities for bond percolation on the lattice $\mathbb{Z}^d$ and the hypercube $Q_d$. These expansions have been rigorously established by Hara and Slade in 1995, and van der Hofstad and Slade in 2006, respectively. We stress that our method does not constitute a branching process proof for the percolation threshold. The analysis of the graph process survival is considerably more challenging than for branching processes in discrete time, due to the interdependence between the descendants of different individuals in the same generation. In fact, it is left open whether the survival probability of $\mathcal{G}(p,q)$ is monotone in $p$ or $q$; we discuss this and some other open problems regarding the new graph process.