> BUC-XIV: 20.08.18 - 24.08.18

Course

Title and Abstract

A

Slides day 1

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Branching Distributional Equations and their Applications

This course will be centered around the theory and applications of distributional equations of the form:

where the are i.i.d. copies of , independent of the vector , and

is a deterministic map. Such equations arise in a wide range of settings, ranging from the analysis of computer algorithms (e.g. Quicksort and PageRank) and the study of queueing networks with synchronization, to applications in biology and statistical physics. Although the solutions to these equations are not in general unique, we are often interested in one particular solution, the so-called attracting endogenous solution. These special solutions can be constructed on a mathematical structure known as a weighted branching process, and their asymptotic behavior can be explicitly computed in many important cases. To complement the theory, we will also introduce a Monte Carlo algorithm based on the bootstrap that can be effectively used for most choices of

In terms of applications, the course will focus on two specific cases, the linear equation (known as the smoothing transform) and the maximum equation (known as the high-order Lindley equation). The former appears in the analysis of Quicksort and Google’s PageRank, while the latter in the analysis of queueing networks with synchronization. The PageRank example in particular will allow us to explore in more detail the connection between branching distributional equations and complex networks. In particular, we will discuss two different types of random graph models that, due to their local tree-like structure, allow us to make the connection between the graph and a (weighted) branching process precise. The coupling techniques involved lie at the core of much or random graph theory, and are therefore of independent interest.

B

Andreas Kyprianou (University of Bath)

Slides

Self-similar Markov processes

We will will reconsider the class of stable processes treated as a self-similar Markov process. This means examining their paths through thte lens through two fundamental transformations: (1) The Lamperti Transformation (2) The Riesz-Bogdan-Zak transfomration. We show how the classical theory of Levy processes feeds in an efficeint way into the theory of self-simiar Markov processes more generally.

C

Elie Aidekon (Paris VI) 

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From branching random walks to nested conformal loops

We will see how we can use branching random walk technics to study extreme nesting in the setting of collections of nested loops in the plane which are conformally invariant. The motivation of such a work comes from the construction of the planar Gaussian free field via CLE_4 by Miller and Sheffield. It gives an alternative construction of the exponential of the Gaussian free field (see Aru, Powell, Sepulveda). It will be based on the paper « The extremal process for nested conformal loops » available at http://www.lpma-paris.fr/dw/doku.php?id=users:[aidekon]:index.

D

Cécile Mailler(Bath) 

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Pólya urns, mutli-type branching processes and smoothing equations.

In this mini-course, I will survey a few results about P&oaccent;loya urns; starting with the almost sure convergence of the original urn of Pólya and Eggenberger, the course will then focus on ``irreducible'' Pólya urns. We will use martingales to show almost sure convergence of the composition of the urn when time tends to infinity, show that Pólya urns can be embedded into continuous time and that their embeddings are multi-type branching processes, and show how smoothing equations can be used to study their asymptotic behaviour, both in discrete and continuous time.

Monday

08:30-09:30

Breakfast

09:30-10:30

A

10:30-11:30

B

11:30-12:00

Coffee

12:00-13:00

C

13:00-14:30

Lunch

14:30-15:30

Arno Siri-Jegousse

Simple nested coalescents

Notes

15:30-18:00

Work sessions with coffee

18:00-20:00

Welcome on the balcony