Igor Kortchemski

Invariance principles for Galton-Watson trees conditioned on the number of leaves

Abstract We are interested in the asymptotic behavior of critical Galton–Watson trees whose offspring distribution may have infinite variance, which are conditioned on having a large fixed number of leaves. We first find an asymptotic estimate for the probability of a Galton–Watson tree having n leaves. Second, we let t n be a critical Galton–Watson tree whose offspring distribution is in the domain of attraction of a stable law, and conditioned on having exactly n leaves. We show that the rescaled Lukasiewicz path and contour function of t n converge respectively to X exc and H exc , where X exc is the normalized excursion of a strictly stable spectrally positive Levy process and H exc is its associated continuous-time height function. As an application, we investigate the distribution of the maximum degree in a critical Galton–Watson tree conditioned on having a large number of leaves. We also explain how these results can be generalized to the case of Galton–Watson trees which are conditioned on having a large fixed number of vertices with degree in a given set, thus extending results obtained by Aldous, Duquesne and Rizzolo.

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 θ ∈ (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 Levy process of index θ. 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.

Limit theorems for conditioned non-generic Galton–Watson trees

We study a particular type of subcritical Galton--Watson trees, which are called non-generic trees in the physics community. In contrast with the critical or supercritical case, it is known that condensation appears in certain large conditioned non-generic trees, meaning that with high probability there exists a unique vertex with macroscopic degree comparable to the total size of the tree. Using recent results concerning subexponential distributions, we investigate this phenomenon by studying scaling limits of such trees and show that the situation is completely different from the critical case. In particular, the height of such trees grows logarithmically in their size. We also study fluctuations around the condensation vertex.

Random stable laminations of the disk

We study large random dissections of polygons. We consider random dissections of a regular polygon with n sides, which are chosen according to Boltzmann weights in the domain of attraction of a stable law of index 2 (1;2]. As n goes to infinity, we prove that these random dissections converge in distribution towards a random compact set, called the random stable lamination. If = 2, we recover Aldous’ Brownian triangulation. However, if 2 (1;2), large faces remain in the limit and a dierent random compact set appears. We show that the random stable lamination can be coded by the continuous-time height function associated to the normalized excursion of a strictly stable spectrally positive Levy process of index . Using this coding, we establish that the Hausdor dimension of the stable random lamination is almost surely 2 1= .

Self-similar scaling limits of Markov chains on the positive integers

We are interested in the asymptotic behavior of Markov chains on the set of positive integers for which, loosely speaking, large jumps are rare and occur at a rate that behaves like a negative power of the current state, and such that small positive and negative steps of the chain roughly compensate each other. If

Simply generated non-crossing partitions

X_{n}

Martingales in self-similar growth-fragmentations and their connections with random planar maps

is such a Markov chain started at

Random stable looptrees

n

Percolation on random triangulations and stable looptrees

, we establish a limit theorem for

The CRT is the scaling limit of random dissections

\frac{1}{n}X_{n}