Improved bounds for 1-independent percolation on Zn

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 S1 and S2, the states of the edges in S1 are independent of the states of the edges in S2. 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 pmax(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 pmax(G) when G is the lattice graph Z2. Since pmax(Zn)≤pmax(Zn−1), it is also of interest to establish the value of limn→∞pmax(Zn). In this paper we significantly improve the best known upper bound on this limit and obtain better upper and lower bounds on pmax(Z2). 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.