Many enumeration problems in combinatorics, including such fundamental questions as the number of regular graphs, can be expressed as high-dimensional complex integrals. Motivated by the need for a systematic study of the asymptotic behavior of such integrals, we establish explicit bounds on the exponentials of complex martingales. Those bounds applied to the case of truncated normal distributions are precise enough to include and extend many enumerative results of Barvinok, Canfield, Gao, Greenhill, Hartigan, Isaev, McKay, Wang, Wormald, and others. Our method applies to sums as well as integrals. As a first illustration of the power of our theory, we considerably strengthen existing results on the relationship between random graphs or bipartite graphs with specified degrees and the so-called β-model of random graphs with independent edges, which is equivalent to the Rasch model in the bipartite case.
- asymptotic enumeration
- complex martingale
- degree sequence
- multidimensional Laplace integral
- random graph