Complex martingales and asymptotic 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. 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.Comment: Minor changes as accepted by Random Structures and Algorithm

Intercalates and discrepancy in random Latin squares

“…To prove Lemma 16, we will use the following estimate due to Ordentlich and Roth [, Proposition 2.2].Theorem For any d (potentially depending on n), let . The number of d‐regular bipartite graphs on is at least We remark that a precise asymptotic estimate for has very recently become available, due to Liebenau and Wormald (the same result was also independently announced by Isaev and McKay ).Proof of Lemma The probability has exactly edges is (here we used Stirling's approximation). By symmetry, each graph with dn edges is equally likely.…”

Sandwiching random regular graphs between binomial random graphs

“…In other words, the conditional distribution of S (λ jk ) with respect to given t is uniform. The random model of S (λ jk ) is referred as the β-model and it is a special case of the exponential family of random graphs, see [7,18] for more details. A further connection between S (λ jk ) and S t is established in Section 5.3.…”

“…where h(θ) = O(n −1/2+6ε ) uniformly for θ ∈ B 0 . A general theory on the estimation of such integrals was developed in [18]. This theory is based on the second-order approximation of complex martingales.…”

“…Here, we quote the results from [18]that were used in Section 5. For a domain Ω ⊆ R n and a twice continuously differentiable function q : Ω → C, define H(q, Ω) = (h jk ), where h jk = sup x∈Ω ∂ 2 q ∂x j ∂x k (x) .…”

“…Consider any product P of a ≥ 1 factors A and b factors B. Representing P = P 1 AP 2 , we get that [18]). Let c 1 , c 2 , c 3 , ε, ρ 1 , ρ 2 , φ 1 , φ 2 be nonnegative real constants with c 1 , ε > 0.…”