Improved bounds for 1-independent percolation on $\mathbb{Z}^n$

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 of the edges in S 1 are independent of the states of the edges in S 2 . Such a model is said to percolate if the random subgraph has an infinite component with positive probability. In 2012 the first author and Bollobás defined p 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 the value of p max (G) when G is the lattice graph Z 2 . Since p max (Z n ) ≤ p max (Z n−1 ), it is also of interest to establish the value of lim n→∞ p max (Z n ). In this paper we significantly improve the best known upper bound on this limit and obtain better upper and lower bounds on p max (Z 2 ). In proving these results, we also give an upper bound on the critical probability for a 1-independent model on the hypercube graph to contain a giant component asymptotically almost surely.

On connectivity in random graph models with limited dependencies

“…Firstly, subgraph independent distributions have been studied under the name of 1-independent random graphs. More generally, a k-independent graph distribution is a distribution where any two sets of edges E, F are independent if the minimum distance between any vertex incident to an edge in E and any vertex incident to an edge in F is at least k. These k-independent graph distributions have been studied extensively in the context of percolation theory, for example on the infinite integer grid [6][7][8]18]. In [28], Day, Falgas-Ravry, and Hancock consider 1-independent random finite graphs.…”

“…To begin, we color each vertex independently with a random color according to its own distribution. 8 For this fixed coloring, we write N(v, c) for the set of neighbors of v, if v is recolored to the color c. We furthermore write Pr c v [⋅] and E c v [⋅] for probabilities and expectations of events and variables given that we keep the fixed coloring for all vertices other than v, and recolor v to c v with c v drawn randomly according to the color distribution Ω v .…”

“…2(n−1) ≤ e −𝜀 2 n∕32 . 8 We once again assume that we have a finite number of colors. Our proof strategy also works with infinite probability spaces.…”