  Let G_n be a sequence of finite transitive graphs with vertex degree d=d(n)
and |G_n|=n. Denote by p^t(v,v) the return probability after t steps of the
non-backtracking random walk on G_n. We show that if p^t(v,v) has quasi-random
properties, then critical bond-percolation on G_n has a scaling window of width
n^{-1/3}, as it would on a random graph.
  A consequence of our theorems is that if G_n is a transitive expander family
with girth at least (2/3 + eps) \log_{d-1} n, then the size of the largest
component in p-bond-percolation with p={1 +O(n^{-1/3}) \over d-1} is roughly
n^{2/3}. In particular, bond-percolation on the celebrated Ramanujan graph
constructed by Lubotzky, Phillips and Sarnak has the above scaling window. This
provides the first examples of quasi-random graphs behaving like random graphs
with respect to critical bond-percolation.
