Existence of a Percolation Threshold on Finite Transitive Graphs

DOI10.1093/IMRN/RNAD222arXiv2208.09501OpenAlexW4386946839MaRDI QIDQ6182622

Publication date: 25 January 2024

Published in: Unnamed Author (Search for Journal in Brave)

Abstract: Let

(G_{n})

be a sequence of finite connected vertex-transitive graphs with volume tending to infinity. We say that a sequence of parameters

(p_{n})

is a percolation threshold if for every

v a r e p s i l o n > 0

, the proportion

l e f t l V e r t K_{1} i g h t V e r t

of vertices contained in the largest cluster under bond percolation

m a t h b b P_{p}^{G}

satisfies both [ �egin{split} lim_{n o infty} mathbb{P}_{(1+varepsilon)p_n}^{G_n} left( leftlVert K_1 ight Vert geq alpha ight) &= 1 quad ext{for some

a l p h a > 0

, and}\ lim_{n o infty} mathbb{P}_{(1-varepsilon)p_n}^{G_n} left( leftlVert K_1 ight Vert geq alpha ight) &= 0 quad ext{for all

a l p h a > 0

}. end{split}] We prove that

(G_{n})

has a percolation threshold if and only if

(G_{n})

does not contain a particular infinite collection of pathological subsequences of dense graphs. Our argument uses an adaptation of Vanneuville's new proof of the sharpness of the phase transition for infinite graphs via couplings [Van22] together with our recent work with Hutchcroft on the uniqueness of the giant cluster [EH21].

Full work available at URL: https://arxiv.org/abs/2208.09501

zbMATH Keywords

Bernoulli bond percolation vertex-transitive gfgraphs

Mathematics Subject Classification ID

Random graphs (graph-theoretic aspects) (05C80) Interacting random processes; statistical mechanics type models; percolation theory (60K35) Combinatorial probability (60C05) Graphs and abstract algebra (groups, rings, fields, etc.) (05C25) Percolation (82B43) Probabilistic methods in extremal combinatorics, including polynomial methods (combinatorial Nullstellensatz, etc.) (05D40)