Components, large and small, are as they should be I: Supercritical percolation on regular graphs of growing degree

We provide sufficient conditions for a regular graph G G of growing degree d d , guaranteeing a phase transition in its random subgraph G p G_p similar to that of G ( n , p ) G(n,p) when p ⋅ d ≈ 1 p· d≈ 1 . These conditions capture several well-studied graphs, such as (percolation on) the complete graph K n K_n , the binary hypercube Q d Q^d , d d -regular expanders, and random d d -regular graphs. In particular, this serves as a unified proof for these (and other) cases. Suppose that G G is a d d -regular graph on n n vertices, with d = ω ( 1 ) d=ω (1) . Let ϵ > 0 ε >0 be a small constant, and let p = 1 + ϵ