Complex martingales and asymptotic enumeration

TL;DR

This paper develops explicit bounds on complex martingales to analyze the asymptotic behavior of high-dimensional integrals in combinatorics, extending enumerative results and strengthening connections between random graph models.

Contribution

It introduces a systematic method for bounding complex martingales, enabling precise asymptotic analysis of combinatorial enumeration problems and random graph models.

Findings

Extended enumerative results for regular graphs and related structures.

Strengthened the relationship between degree-constrained graphs and the β-model.

Provided bounds applicable to sums and integrals in high-dimensional settings.

Abstract

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 behaviour 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 $\beta$-model of random graphs with independent edges, which is equivalent to the Rasch model in the bipartite case.