Long paths and connectivity in {$1$}-independent random graphs

TL;DR

This paper investigates connectivity and long path properties in 1-independent percolation models on graphs, providing new bounds on critical probabilities and exact results for specific finite graphs, extending classical percolation theory.

Contribution

It introduces improved bounds on the critical probability for infinite clusters in 1-independent models and determines exact connectivity probabilities for certain finite graphs.

Findings

New lower bounds on the critical probability p* for infinite clusters.

Exact formulas for connectivity probabilities in paths, cycles, and complete graphs.

Determination of the 1-independent critical probability for long paths on line and ladder lattices.

Abstract

Given a graph $G$, a probability measure $\mu$ on the subsets of the edge set of $G$ is said to be $1$-independent if events determined by edge sets that are at graph distance at least $1$ apart in $G$ are independent. Call such a probability measure a $1$-ipm on $G$, and denote by $\mathbf{G}_{\mu}$ the associated random spanning subgraph of $G$. Let $\mathcal{M}_{1,\geqslant p}(G)$ (resp. $\mathcal{M}_{1,\leqslant p}(G)$) denote the collection of $1$-ipms $\mu$ on $G$ for which each edge is included in $\mathbf{G}_{\mu}$ with probability at least $p$ (resp. at most $p$). Let $\mathbb{Z}^2$ denote the square integer lattice. Balister and Bollob\'as raised the question of determining the critical value $p_{\star}=p_{1,c}(\mathbb{Z}^2)$ such that for all $p>p_{\star}$ and all $\mu \in \mathcal{M}_{1,\geqslant p}(\mathbb{Z}^2)$, $\left(\mathbf{\mathbb{Z}^2}\right)_{\mu}$ almost surely contains an infinite component. This can be thought of as asking for a $1$-independent analogue of the celebrated Harris--Kesten theorem. In this paper we investigate both this problem and connectivity problems for $1$-ipms more generally. We give two lower bounds on $p_{\star}$ that significantly improve on the previous bounds. Furthermore, motivated by the Russo--Seymour--Welsh lemmas, we define a $1$-independent critical probability for long paths and determine its value for the line and ladder lattices. Finally, for finite graphs $G$ we study $f_{1,G}(p)$ (respectively $F_{1,G}(p)$), the infimum (resp. supremum) over all $\mu\in \mathcal{M}_{1,\geqslant p}(G)$ (resp. all $\mu \in \mathcal{M}_{1,\leqslant p}(G)$) of the probability that $\mathbf{G}_{\mu}$ is connected. We determine $f_{1,G}(p)$ and $F_{1,G}(p)$ exactly when $G$ is a path, a complete graph and a cycle of length at most $5$. Many new problems arise from our work, which are discussed in the final section of the paper.