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


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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