Percolation on a Feynman Diagram

In a recent paper [1] we investigated Potts models on “thin” random graphs – generic Feynman diagrams, using the idea that such models may be expressed as the N → 1 limit of a matrix model. The models displayed first order transitions for all q > 2, giving identical behaviour to the corresponding Bethe lattice. We use here one of the results of [1] namely a general saddle point solution for a q state Potts model expressed as a function of q, to investigate some peculiar features of the percolative limit q → 1 and compare the results with those on the Bethe lattice.