Long paths and connectivity in 1‐independent random graphs

Abstract

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 G the associated random graph model. Let  1,⩾p (G) denote the collection of 1-ipms on G for which each edge is included in G 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 ∈  1,⩾p (G), (G) 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 ∈  1,⩾p (G) of the probability that G 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.

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

“…In this model, each edge is open with probability p 4 + (1 − p) = 1 − 3 4 p, so p max (Z 2 ) ≥ 1 − 3 4 p site . Now, as noted in the introduction, Day, Falgas-Ravry, and Hancock [10] showed that…”

“…The best known lower bound on both p max (Z 2 ) and lim n→∞ p max (Z n ) comes from a construction of Day, Falgas-Ravry, and Hancock in [10] which gives…”

“…It was conjectured by Falgas-Ravry and Pfenninger in [12] that there exists some n ≥ 3 for which this is an equality. In [10] the authors also give a separate construction which yields p max (Z 2 ) ≥ p 2 site + 1 2 (1 − p site ), where p site = p site (Z 2 ) denotes the critical probability for independent site percolation on Z 2 , that is, the supremum of the p such that percolation does not occur in the percolation model on Z 2 in which vertices are open independently with probability p and edges are open if both their endpoints are open. The best known rigorous lower bound on p site is 0.556 due to van den Berg and Ermakov [20].…”

“…In this section we discuss some interesting related problems on 1-independent percolation models on hypercubes and lattices. We begin by restating a question first posed by Day, Falgas-Ravry, and Hancock in [10].…”

“…The value of p max (G) for different infinite graphs G has been studied, along with various other properties of 1-independent bond percolation models, in [2,3,10,12,16]. In particular, in [2] the authors determine p max (T ) for all locally finite infinite trees T in terms of a parameter known as the branching number of T .…”