Components, Large and Small, Are as They Should Be II: Supercritical Percolation on Regular Graphs of Constant Degree

Let d ≥ 3 be a fixed integer. Let y := y(p) be the probability that the root of an infinite d-regular tree belongs to an infinite cluster after p-bond-percolation. We show that for all constants b, α > 0 and 1 < λ < d − 1, there exist constants c, C > 0 such that the following holds. Let G be a d-regular graph on n vertices, satisfying that for every U ⊆ V(G) with |U| ≤ ⁿ₂ , e(U, U^c) ≥ b|U| and for every U ⊆ V(G) with |U| ≤ log^C n, e(U) ≤ (1 + c)|U|. Let p = _d−^λ₁ . Then, with probability tending to one as n tends to infinity, the largest component L1 in the random subgraph Gp of G satisfies |||1 − ^|_yn^L1| ||| ≤ α, and all the other components in Gp are of order O ( _(λ^λlog₋₁₎ⁿ₂ ) . This generalises (and improves upon) results for random d-regular graphs.