Long paths and connectivity in 1-independent random graphs

A probability measure $$ on the subsets of the edge set of a graph G is a 1-independent probability measure (1-ipm) on G if events determined by edge sets that are at graph distance at least 1 apart in G are independent. Given a 1-ipm $$ , denote by $$ the associated random graph model. Let $$ denote the collection of 1-ipms $$ on G for which each edge is included in $$ with probability at least p. For $$ , Balister and Bollobás asked for the value of the least p _⋆ such that for all p > p _⋆ and all $$ , $$ almost surely contains an infinite component. In this paper, we significantly improve previous lower bounds on p _⋆. We also determine the 1-independent critical probability for the emergence of long paths on the line and ladder lattices. Finally, for finite graphs G we study f _1, (p), the infimum over all $$ of the probability that $$ is connected. We determine f _1, (p) exactly when G is a path, a complete graph and a cycle of length at most 5.