Paul Balister and Michael Savery (University of Oxford) : Improved bounds for 1-independent percolation on $Z^n$

Abstract:

A 1-independent bond percolation model on a graph $G$ is a probability distribution on the spanning subgraphs of $G$ in which, for all vertex-disjoint sets of edges $S_1$ and $S_2$ , the states (i.e. present or not present) of the edges in $S_1$ are independent of the states of the edges in $S_2$ . Such models typically arise in renormalisation arguments applied to independent percolation models, or percolation models with finite range dependencies. A 1-independent model is said to percolate if the random subgraph has an infinite component with positive probability. In 2012 Balister and Bollob\’as defined $p_{\mathrm{max}}(G)$ to be the supremum of those $p$ for which there exists a 1-independent bond percolation model on $G$ in which each edge is present in the random subgraph with probability at least $p$ but which does not percolate. A fundamental and challenging problem in this area is to determine, or give good bounds on, the value of $p_{\mathrm{max}}(G)$ when $G$ is the lattice graph $\mathbb{Z}^2$ . Since $p_{\mathrm{max}}(\mathbb{Z}^n)\leq p_{\mathrm{max}}(\mathbb{Z}^{n-1})$ , it is also of interest to establish the value of $\lim_{n\to\infty}p_{\mathrm{max}}(\mathbb{Z}^n)$ .

In this talk we will present a significantly improved upper bound for this limit as well as improved upper and lower bounds for $p_{\mathrm{max}}(\mathbb{Z}^2)$ . We will also show that with high confidence we have $p_{\mathrm{max}}(\mathbb{Z}^n)<p_{\mathrm{max}}(\mathbb{Z}^2)$ for large $n$ and discuss some open problems concerning 1-independent models on other graphs.

This is joint work with Tom Johnston and Alex Scott.

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Based on

https://arxiv.org/abs/2206.12335

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Paul Balister and Michael Savery (University of Oxford) : Improved bounds for 1-independent percolation on $Z^n$

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