Complex martingales and combinatorial enumeration

Abstract: 

Many enumeration problems in combinatorics, including such fundamental questions as the number of regular graphs, can be expressed as high-dimensional complex integrals. The asymptotic behaviour is found by concentrating the integral in a small region and then approximating the integrand inside that region.

Evaluation of such integrals one variable at a time produces a martingale and the sought-after quantity has the form Eef(X)Eef(X), where f(X)f(X) is some complex-valued function of a multidimensional measure (usually a truncated gaussian measure).

This talk is based on joint work with B.D. McKay and C. Greenhill, where we generalise the problem to arbitrary complex

martingales and find conditions under which Eef∼eEf+12VfEef∼eEf+12Vf, where VfVf is the  pseudovariance.

The result appears to cover all previous combinatorial enumeration  problems done by the integral method, and also applies to discrete martingales such as functions of random permutations and random subsets.