A considerable number of asymptotic distributions arising in random combinatorics and analysis of algorithms are of the exponential-quadratic type, that is, Gaussian. We exhibit a class of "universal" phenomena that are of the exponential-cubic type, corresponding to distributions that involve the Airy function. In this paper, such Airy phenomena are related to the coalescence of saddle points and the confluence of singularities of generating functions. For about a dozen types of random planar maps, a common Airy distribution (equivalently, a stable law of exponent 3/2) describes the sizes of cores and of largest (multi)connected components. Consequences include the analysis and fine optimization of random generation algorithms for multiply connected planar graphs. Based on an extension of the singularity analysis framework suggested by the Airy case, the paper also presents a general classification of compositional schemas in analytic combinatorics.

Abstract. We present a simple encoding of plane triangulations (aka. maximal planar graphs) by plane trees with two leaves per inner node. Our encoding is a bijection taking advantage of the minimal Schnyder tree decomposition of a plane triangulation. Coding and decoding take linear time. As a byproduct we derive: (i) a simple interpretation of the formula for the number of plane triangulations with n vertices, (ii) a linear random sampling algorithm, (iii) an explicit and simple information theory optimal encoding. 1

We propose a new linear time algorithm to represent a planar graph. Based on a specific triangulation of the graph, our coding takes on average 5.03 bits per node, and 3.37 bits per node if the graph is maximal. We derive from this representation that the number of unlabeled planar graphs with n nodes is at most 2 n+O(log n) where 5.007. The current lower bound is 2 n+(log n) for 4.71. We also show that almost all unlabeled and almost all labeled n-node planar graphs have at least 1.70n edges and at most 2.54n edges.

We present an expected polynomial time algorithm to generate a labeled planar graph uniformly at random. To generate the planar graphs, we derive recurrence formulas that count all such graphs with n vertices and m edges, based on a decomposition into 1-, 2-, and 3-connected components. For 3-connected graphs we apply a recent random generation algorithm by Schaeffer and a counting formula by Mullin and Schellenberg.

. The Airy distribution (of the \area" type) occurs as limit distribution of cumulative parameters in a number of combinatorial structures, like path length in trees, area below walks, displacement in parking sequences, and it is also related to basic graph and polyomino enumeration. We obtain curious explicit evaluations for certain moments of the Airy distribution, including moments of orders 1; 3; 5; &c, as well as + 1 3 ; 5 3 ; 11 3 ; &c. and 7 3 ; 13 3 ; 19 3 ; &c . Our proofs are based on integral transforms of the Laplace and Mellin type and they rely essentially on \non-probabilistic" arguments like analytic continuation. A by-product of this approach is the existence of relations between moments of the Airy distribution, the asymptotic expansion of the Airy function Ai(z) at +1, and power symmetric functions of the zeros k of Ai(z). For probabilists, the Airy distribution considered here is nothing but the distribution of the area under the Brownian excursion. The ...

Asymptotic formulae for two-dimensional arrays (fr,s)r,s≥0 where the associated generating function F (z, w): = � fr,szrw s is meromorphic are provided. Our ap-r,s≥0 proach is geometrical. To a big extent it generalizes and completes the asymptotic description of the coefficients fr,s along a compact set of directions specified by smooth points of the singular variety of the denominator of F (z, w). The scheme we develop can lead to a high level of complexity. However, it provides the leading asymptotic order of fr,s if some unusual and pathological behavior is ruled out. It relies on the asymptotic analysis of a certain type of stationary phase integral of the form � e −s·P (d,θ) A(d, θ)dθ, which describes up to an exponential factor the asymptotic be-havior of the coefficients fr,s along the direction d = r s in the (r, s)-lattice. The cases of interest are when either the phase term P (d, θ) or the amplitude term A(d, θ) exhibits a change of degree as d approaches a degenerate direction. These are han-dled by a generalized version of the stationary phase and the coalescing saddle point method which we propose as part of this dissertation. The occurrence of two spe-cial functions related to the Airy function is established when two simple saddles of the phase term coalesce. A scheme to study the asymptotic behavior of big powers of generating functions is proposed as an additional application of these generalized methods. ii Dedicated to my mother, father and sister. iii ACKNOWLEDGMENTS I would like to thank to my advisor, Robin Pemantle, for his support and guidance throughout my graduate years at Ohio State. I am deeply indebted for he having supported me as his research assistant for an extended period of time. I also offer my gratitude for his unconditional commitment to connect and keep me in touch with

Given a multivariate generating function F (z1 ; : : : ; zd ) = P ar 1 ;:::;r d z r 1 1 z r d d , we determine asymptotics for the coecients. Our approach is to use Cauchy's integral formula near singular points of F , resulting in a tractable oscillating integral. This paper treats the case where the singular point of F is a smooth point of a surface of poles. Companion papers G treat singular points of F where the local geometry is more complicated, and for which other methods of analysis are not known.

Abstract. Uniform asymptotic formulae for arrays of complex numbers of the form (fr,s), with r and s nonnegative integers, are provided as r and s converge to infinity at a comparable rate. Our analysis is restricted to the case in which the generating function F(z, w): = P fr,sz r w s is meromorphic in a neighborhood of the origin. We provide uniform asymptotic formulae for the coefficients fr,s along directions in the (r,s)-lattice determined by regular points of the singular variety of F. Our main result derives from the analysis of a one dimensional parameter-varying integral describing the asymptotic behavior of fr,s. We specifically consider the case in which the phase term of this integral has a unique stationary point, however, allowing the possibility that one or more stationary points of the amplitude term coalesce with this. Our results find direct application in certain problems associated to the Lagrange inversion formula as well as bivariate generating functions of the form v(z)/(1 − w · u(z)). 1.

We propose a new linear time algorithm to represent a planar graph. Based on a speci c triangulation of the graph, our coding takes on average 5:03 bits per node, and 3:37 bits per node if the graph is maximal. We derive from this representation that the number of unlabeled planar graphs with n nodes is at most 2 n+O(log n) , where 5:007.

Abstract. Given an integer m ≥ 1, let ‖· ‖ be a norm in R m+1 and let S m + denote the set of points d = (d0,..., dm) in R m+1 with nonnegative coordinates and such that ‖d ‖ = 1. Consider for each 1 ≤ j ≤ m a function fj(z) that is analytic in an open neighborhood of the point z = 0 in the complex plane and with possibly negative Taylor coefficients. Given n = (n0,..., nm) in Z m+1 with nonnegative coordinates, we develop a method to systematically associate a parameter-varying integral to study the asymptotic behavior of the coefficient of z n0 of the Taylor series of Q m j=1 {fj(z)}n j, as ‖n ‖ → ∞. The associated parametervarying integral has a phase term with well specified properties that make the asymptotic analysis of the integral amenable to saddle-point methods: for many d ∈ S m +, these methods ensure uniform asymptotic expansions for [z n0 Q