Multinomial percolation

We look at bond percolation on “graph blowups” of a graph G. This is a generalization of the Erdos-Renyi random graph, and it features an analogous phase transition with respect to the appearance of a giant component. We show that the vector multiplicities of the giant component converge (after suitably centering and rescaling) to a Gaussian field on G whose covariance can be computed explicitly as the square of a massive Green’s function on G. The proof strategy is combinatorial and relies on the combinatorics of spanning trees on blowup graphs, whose generating function has a beautiful analytic structure.