Uniform bounds for exponential moment of maximum of a Dyck paths

Let D be a Dyck path chosen uniformly from the set of Dyck paths with 2n steps. We prove that the sequence of the exponential moments of the maximum of D normalized by the square root of n converges in the limit of infinite n, and therefore is bounded uniformly in n. This result justifies corresponding assumption used to prove certain estimates of high moments of large random matrices.Comment: 7 pages; accepted by Electronic Communications in Probabilit

A simple proof of Duquesne's theorem on contour processes of conditioned Galton-Watson trees

We give a simple new proof of a theorem of Duquesne, stating that the properly rescaled contour function of a critical aperiodic Galton-Watson tree, whose offspring distribution is in the domain of attraction of a stable law of index $\theta \in (1,2]$, conditioned on having total progeny $n$, converges in the functional sense to the normalized excursion of the continuous-time height function of a strictly stable spectrally positive L\'evy process of index $\theta$. To this end, we generalize an idea of Le Gall which consists in using an absolute continuity relation between the conditional probability of having total progeny exactly $n$ and the conditional probability of having total progeny at least $n$. This new method is robust and can be adapted to establish invariance theorems for Galton-Watson trees having $n$ vertices whose degrees are prescribed to belong to a fixed subset of the positive integers.Comment: 16 pages, 2 figures. Published versio

The topological structure of scaling limits of large planar maps

We discuss scaling limits of large bipartite planar maps. If p is a fixed integer strictly greater than 1, we consider a random planar map M(n) which is uniformly distributed over the set of all 2p-angulations with n faces. Then, at least along a suitable subsequence, the metric space M(n) equipped with the graph distance rescaled by the factor n to the power -1/4 converges in distribution as n tends to infinity towards a limiting random compact metric space, in the sense of the Gromov-Hausdorff distance. We prove that the topology of the limiting space is uniquely determined independently of p, and that this space can be obtained as the quotient of the Continuum Random Tree for an equivalence relation which is defined from Brownian labels attached to the vertices. We also verify that the Hausdorff dimension of the limit is almost surely equal to 4.Comment: 45 pages Second version with minor modification

Ideals In A Forest, One-Way Infinite Binary Trees And The Contraction Method

The analysis of an algorithm by Koda and Ruskey for listing ideals in a forest poset leads to a study of random binary trees and their limits as infinite random binary trees. The corresponding finite and infinite random forests are studied too. The infinite random binary trees and forests studied here have exactly one infinite path; they can be defined using suitable size-biazed Galton--Watson processs. Limit theorems are proved using a version of the contraction method