Some measure theory on stacks of graphs

Abstract

We apply a theorem of Wick to rewrite certain classes of exponential measures on random graphs as integrals of Feynman-Gibbs type, on the real line. The analytic properties of these measures can then be studied in terms of phase transitions; spaces of scale-free trees are a particularly interesting example.

One background technical issue is that graphs up to isomorphism are a kind of moduli space, which has orbifold points, whose symmetries make counting delicate. Revised, to correct a reprehensible sign error; references updated