Exchangeable pairs, switchings, and random regular graphs

We consider the distribution of cycle counts in a random regular graph, which is closely linked to the graph's spectral properties. We broaden the asymptotic regime in which the cycle counts are known to be approximately Poisson, and we give an explicit bound in total variation distance for the approximation. Using this result, we calculate limiting distributions of linear eigenvalue functionals for random regular graphs. Previous results on the distribution of cycle counts by McKay, Wormald, and Wysocka (2004) used the method of switchings, a combinatorial technique for asymptotic enumeration. Our proof uses Stein's method of exchangeable pairs and demonstrates an interesting connection between the two techniques.Comment: Very minor changes; 23 page

Bethe approximation for self-interacting lattice trees

In this paper we develop a Bethe approximation, based on the cluster variation method, which is apt to study lattice models of branched polymers. We show that the method is extremely accurate in cases where exact results are known as, for instance, in the enumeration of spanning trees. Moreover, the expressions we obtain for the asymptotic number of spanning trees and lattice trees on a graph coincide with analogous expressions derived through different approaches. We study the phase diagram of lattice trees with nearest-neighbour attraction and branching energies. We find a collapse transition at a tricritical theta point, which separates an expanded phase from a compact phase. We compare our results for the theta transition in two and three dimensions with available numerical estimates.Comment: 10 pages, 3 figures, to be published in Europhysics Letter

A Complete Grammar for Decomposing a Family of Graphs into 3-connected Components

Tutte has described in the book "Connectivity in graphs" a canonical decomposition of any graph into 3-connected components. In this article we translate (using the language of symbolic combinatorics) Tutte's decomposition into a general grammar expressing any family of graphs (with some stability conditions) in terms of the 3-connected subfamily. A key ingredient we use is an extension of the so-called dissymmetry theorem, which yields negative signs in the grammar. As a main application we recover in a purely combinatorial way the analytic expression found by Gim\'enez and Noy for the series counting labelled planar graphs (such an expression is crucial to do asymptotic enumeration and to obtain limit laws of various parameters on random planar graphs). Besides the grammar, an important ingredient of our method is a recent bijective construction of planar maps by Bouttier, Di Francesco and Guitter.Comment: 39 page

Asymptotic Formulae for Restricted Unimodal Sequences

Additive enumeration problems, such as counting the number of integer partitions, lie at the intersection of various branches of mathematics including combinatorics, number theory, and analysis. Extending partitions to integer unimodal sequences has also yielded interesting combinatorial results and asymptotic formulae, which form the subject of this thesis. Much like the important work of Hardy and Ramanujan proving the asymptotic formula for the partition function, Auluck and Wright gave similar formulas for unimodal sequences. Following the circle method of Wright, we provide the asymptotic expansion for unimodal sequences with odd parts. This is then generalized to a two-parameter family of mixed congruence relations, with parts on one side with parts on one side up to the peak satisfying r (mod m) and parts on the other side -r (mod m), and an asymptotic formula is provided. Techniques used in the proofs include Wright\u27s circle method, modular transformations, and bounding of complex integrals

The Combinatorics of Alternating Tangles: from theory to computerized enumeration

We study the enumeration of alternating links and tangles, considered up to topological (flype) equivalences. A weight $n$ is given to each connected component, and in particular the limit $n\to 0$ yields information about (alternating) knots. Using a finite renormalization scheme for an associated matrix model, we first reduce the task to that of enumerating planar tetravalent diagrams with two types of vertices (self-intersections and tangencies), where now the subtle issue of topological equivalences has been eliminated. The number of such diagrams with $p$ vertices scales as $12^p$ for $p\to\infty$. We next show how to efficiently enumerate these diagrams (in time $\sim 2.7^p$) by using a transfer matrix method. We give results for various generating functions up to 22 crossings. We then comment on their large-order asymptotic behavior.Comment: proceedings European Summer School St-Petersburg 200