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 by Gμ the associated random graph model. Let ℳ1,⩾p(G) denote the collection of 1‐ipms μ on G for which each edge is included in Gμ with probability at least p. For G=Z2, Balister and Bollobás asked for the value of the least p ⋆ such that for all p > p ⋆ and all μ∈ℳ1,⩾p(G), Gμ 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∈ℳ1,⩾p(G) of the probability that Gμ 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.

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