UK Easter Probability Meeting 2018

A workshop about:
Random Dynamics and Other Recent Developments

April 9th-13th, University Of Sheffield

Each minicourse will consist of four 1h sessions, including time for open discussion.

Around the Brownian Continuum Random Tree
Christina Goldschmidt (University of Oxford)
My aim in this course is to give first an introduction to the Brownian continuum random tree, which is the scaling limit of a broad family of random discrete trees. I will then give an overview of its properties, how to characterise it, and some of the ways it gets used as a building block in other structures. There will be an emphasis on how to do calculations with this a priori rather complicated object. Another theme will be the interplay between the discrete and continuous mathematical objects which arise.

Slides used for the minicourse.

Some relevant literature:

  1. D. Aldous, The Brownian continuum random tree II: An overview
  2. J.-F. Le Gall, Random trees and applications
  3. J. Pitman, Combinatorial stochastic processes

  1. L. Addario-Berry, O. Angel, G. Chapuy, E. Fusy and C. Goldschmidt, Voronoi tessellations in the CRT and continuum random maps of finite excess
  2. L. Addario-Berry, N. Broutin and C. Goldschmidt, The continuum limit of critical random graphs
  3. L. Addario-Berry, N. Broutin and C. Goldschmidt, Critical random graphs: limiting constructions and distributional properties
  4. M. Albenque and C. Goldschmidt, The Brownian continuum random tree as the unique solution to a fixed-point equation

Stochastic Peturbations of Dynamical Systems and Other Topics
Robin Pemantle (University of Pennsylvania)
I will begin by describing the cooled stochastic approximation paradigm, giving classical examples and outlining results and methods. Next I will discuss some recent results on trapping and end with some work in progress. In separate talks, I will discuss some other problems, such as recovery of a message from transmissions through a deletion channel, aspects of percolation on random trees, and a large deviation problem with applications to combinatorics.

  1. 1. Stochastic perturbations of dynamical systems I: the cooled paradigm
  2. 2. Stochastic perturbations of dynamical systems II: case studies
  3. 3. Invasion percolation on Galton-Watson trees
  4. 4. How many transmissions are required in a deletion channel?

Self-Interacting Random Walks and Statistical Physics
Pierre Tarrès (NYU Shanghai and Université Paris-Dauphine)
In this course we explain how the Edge-reinforced random walk, introduced by Coppersmith and Diaconis in 1986, is related to several models in statistical physics, namely the supersymmetric hyperbolic sigma model introduced by Zirnbauer (1991), Disertori, Spencer and Zirnbauer (2010), the random Schrödinger operator and Dynkin's isomorphism.

We start by a short review of self-interacting random walks and introduce a few techniques to build up intuition on the results that can be expected from such walks.

Then we present the timeline construction for generalised Pólya urns and for the self-interacting walk, originally proposed by Davis and Rubin (1990). That technique enables one to show localisation of strongly edge-reinforced random walks. Using a result a Kendall (1966) on the Yule branching process one can offer an alternative description of the case of linear interaction, and show in particular an explicit link between the linearly edge-reinforced random walk and the Vertex-reinforced jump process originally introduced by Davis and Volkov (2004), see Tarrès (2011), Sabot and Tarrès (2015).

Finally we explain how the supersymmetric hyperbolic sigma model and the random Schrödinger operator arises in the description of the limit behavior of the Vertex-reinforced jump process, see Sabot and Tarrès (2015), Sabot, Tarrès and Zeng (2017), Sabot and Zeng (2015).