We study random subgraphs of an arbitrary finite connected transitive graph
G obtained by independently deleting edges with probability 1−p.
Let V be the number of vertices in G, and let Ω be their
degree. We define the critical threshold pc=pc(G,λ) to be the
value of p for which the expected cluster size of a fixed vertex attains the
value λV1/3, where λ is fixed and positive. We show that
for any such model, there is a phase transition at pc analogous to the phase
transition for the random graph, provided that a quantity called the triangle
diagram is sufficiently small at the threshold pc. In particular, we show
that the largest cluster inside a scaling window of size
|p-p_c|=\Theta(\cn^{-1}V^{-1/3}) is of size Θ(V2/3), while below
this scaling window, it is much smaller, of order
O(ϵ−2log(Vϵ3)), with \epsilon=\cn(p_c-p). We also obtain
an upper bound O(\cn(p-p_c)V) for the expected size of the largest cluster
above the window. In addition, we define and analyze the percolation
probability above the window and show that it is of order \Theta(\cn(p-p_c)).
Among the models for which the triangle diagram is small enough to allow us to
draw these conclusions are the random graph, the n-cube and certain Hamming
cubes, as well as the spread-out n-dimensional torus for n>6