Long paths and connectivity in 1-independent random graphs

A probability measure μ 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 μ , denote byG μ the associated random graph model. Letℳ 1 , ⩾ p ( G ) denote the collection of 1-ipms μ on G for which each edge is included inG μ with probability at least p. For G = Z 2 , Balister and Bollobás asked for the value of the least p ⋆ such that for all p > p ⋆ and all μ ∈ ℳ 1 , ⩾ p ( G ) ,G μ 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, G (p), the infimum over all μ ∈ ℳ 1 , ⩾ p ( G ) of the probability thatG μ is connected. We determine f 1, G (p) exactly when G is a path, a complete graph and a cycle of length at most 5.