Existence of a Percolation Threshold on Finite Transitive Graphs

Let (Gₙ) be a sequence of finite connected vertex-transitive graphs with volume tending to infinity. We say that a sequence of parameters (pₙ) is a percolation threshold if for every ε> 0, the proportion K₁ of vertices contained in the largest cluster under bond percolation PₚG satisfies both $$ {align*} {split}{} limn → ∞ {P}_{(1+ε)pₙ}^{Gₙ} ( K₁ ≥ α ) &= 1 for some\ α> 0, and\\ limn → ∞ {P}_{(1-ε)pₙ}^{Gₙ} ( K₁ ≥ α ) &= 0 for all\ α > 0. {split} {align*}$$We prove that $(Gₙ)$ has a percolation threshold if and only if $(Gₙ)$ 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 [27] together with our recent work with Hutchcroft on the uniqueness of the giant cluster [15].