  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.
