Workshop on the occasion of Prof. Alexander Iksanov’s 50th birthday

Presentation

I will speak about some recent results obtained together with Sasha and Dariusz Buraczewski about the recurrence/transience of random affine recursions X_n=M_nX_n-1+Q_n, n=1,2,… for i.i.d. R₊²-valued (M₁,Q₁), (M₂,Q₂),… in cases when the Goldie-Maller conditions for positive recurrence fail to hold. This amounts to a discussion of the critical case when the random walk S_n=log(M₁)+log(M₂)+…+log(M_n) is oscillating, and the divergent contractive case when S_n → -∞ a.s. (negative divergence), but positive recurrence of (X_n)_n≥0 fails to hold.

The integer points of the real line are marked by the positions of a standard random walk with positive integer jumps. We assume that the set of marked sites is strongly sparse, i.e. the jumps of the random walk have infinite mean and regularly varying. We consider a nearest neighbor random walk on the set of integers having jumps +/-1 with probability 1/2 at every nonmarked site, whereas a random drift is imposed at every marked site. We will present some new limit theorems for the so defined random walk in a strongly sparse random environment. The talk will be based on a joint work with Alexander Iksanov, Piotr Dyszewski and Alexander Marynych.

Presentation

A recurring theme of my joint work with Alexander is the fluctuations of martingales in branching processes. I will give a brief overview of our joint works in this area.

A branching random walk is a population model on the real line in which each particle moves and creates offspring around its location in an independent manner. This system invades its environment at a linear growth rate, with a random delay that can be expressed in terms of the limit of the so-called derivative martingale. In this talk, we will study the right tail of this random variable by characterizing the sub-harmonic functions of the killed random walk with linear growth. This is based on a joint work with Dariusz Buraczewski and Alexander Iksanov.

Presentation

40-minutes break

In this talk we shall review limit theorems for profiles of binary search trees and random recursive trees. A simple inductive way to generate a random recursive tree is as follows. At time 1 we start with a tree consisting of a single vertex (the root) labeled with 1. At each time k, given a recursive tree with k vertices, we choose one of the k vertices uniformly at random and add to this vertex an offspring labeled with k+1. The random tree obtained at time n is called the random recursive tree with n vertices. We are interested in the profile of the random recursive tree, which is the function k→L_n(k) counting the number of vertices at distance k from the root in a tree with n vertices. We shall review distributional and functional limit theorems for the random variables L_n(k) as n→∞ and k=k(n) is some function of n (which may stay constant or go to ∞ at some speed). The talk is based on joint works with A. Iksanov, A. Marynych and H. Sulzbach.

A particular coupling related to a random walk allows to derive a weak convergence result for the number of cuts needed to isolate the root of a random recusive tree. This result has direct consequences for the number of collisions that take place in the Bolthausen-Sznitman coalescent until there is just a single block. The coupling turns out to be useful to derive the asymptotics of the absorption time of certain Markov chains, in particular the number of collisions in beta(a,1)-coalescents with parameter 0 < a < 2. Generalizations are briefly discussed. The talk is based on a cute joint work with Alexander Iksanov published in 2007.

Presentation

Many classic combinatorial structures are decomposable in component elements called blocks. A random structure is regenerative if removing in some way a block yields a reduced structure similar to the whole. This general idea has been realised in terms of exchangeable partitions and their ordered counterparts, and connected to continuous time/space processes.

The talk is devoted to Sasha Iksanov’s fundamental contribution to the analysis of this connection and
application of the renewal theory methods to the asymptotic block-counting.

If one aims to single out the most important unifying ingredient of our hero-of-the-day’s works, the most accurate answer should be “(highly advanced) renewal theory”! In this speech I will provide numerous evidences to this statement and bring together under a single umbrella all the previous talks.

Presentation