The expansion of random regular graphs

Our aim is now to show that for any d ≥ 3, almost all d-regular graphs on {1, 2, . . . , n} have edge-expansion ratio at least cdd (if nd is even), where cd > 0 depends only upon d. (In fact, we will see that we can take cd ≥ 0.18 for all d ≥ 3, and cd → 1/2 as d → ∞.) More precisely, if G(n, d) denotes a (uniform) random d-regular graph on [n], meaning a (labelled) d-regular graph on [n] chosen uniformly at random from the set of all d-regular graphs on [n], then P{h(Gn,d) ≥ cdd} → 1 as n→∞.