# Random walks and related random topics

This graduate school will be held by the University of Göttingen from 17.10.2022 to 21.10.2022.

There will be four mini-courses on topics related with random processes and probabilistic constructions, with focus on random walks on spaces, group theory and geometry. The lectures are aimed at non-specialist graduate students, and there will be a gong session of short contributed talks.

Lecturers

Poster

Program

Schedule

Gong Talks

## Lecturers

Adrien Boulanger (Marseille)

John Mackay (Bristol)

Laura Monk (Bristol)

Tatiana Nagnibeda (Genève)

## Poster here

## Program

__Adrien Boulanger - Hyperbolic random walks__

Abstract: The two main themes of this mini-course will be geometric group theory and random walks. The course will begin with a rather general introduction on random walks. We will progressively derive towards random walks on groups and, more specifically, on random walks on hyperbolic groups.

__John Mackay - Random Groups__

Abstract: In this course we'll consider random finitely presented groups. There are a variety of different approaches to such objects, but we will focus on the density models of Gromov, where for a given number of generators and length of relations, the density parameter controls how many relations are chosen at random. These models for random groups exhibit interesting behaviour, often being hyperbolic, and with variation in properties at different densities.

__Laura Monk - Geometry and spectrum of random hyperbolic surfaces__

Abstract: The core focus of this course is the study of compact hyperbolic surfaces, that is to say surfaces of constant curvature -1. We will want to examine their geometry (the length of curves, the diameter, ...) as well as their spectrum (their sounds, if used as a drum). A significant challenge when trying to describe these surfaces is that it is quite easy to exhibit some families of "bad" surfaces, which seem very pathological and ill-behaved. This is where probabilities come into play: we will sample random surfaces and prove properties true with probability close to 1 rather than all of the time. There are several ways to do so, and the model we will use is the Weil-Petersson model, which relies on a beautiful toolbox developed by Mirzakhani. I will introduce this random model and the methods used to study random surfaces sampled this way. This will allow us to get a picture of what a typical surface looks like.

Hand-written notes available under direct request at Laura's mail address.

__Tatiana Nagnibeda - Amenability and probabilities on graphs and groups__

Abstract: Starting with the work of John von Neumann on the Banach-Tarski Paradox, many interesting connections have emerged between the study of infinite (in these lectures, finitely generated) groups on one hand and probability theory, measure theory and functional analysis on the other hand. One example of such connection that we will see in the course is Kesten’s criterion that characterizes amenable groups in terms of asymptotic behaviour of random walks on their Cayley graphs. It turns out to be relevant to the solution of the so-called von Neumann’s Problem – an attempt to give an algebraic characterization of the class of amenable groups.

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## Schedule

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## Gong Talks

**Anna Roig Sanchis**(IMJ-PRG, Sorbonne Université):

*A model for random (hyperbolic) 3-manifolds*

**Antoine Goldsborough**(Heriot-Watt University):

*Random walks and quasi-isometries*

**Cipriana Anghel-Stan**(IMAR (Inst of Math of the Romanian Academy)):

*Heat kernels asymptotics for real powers of Laplacians*

**David Guo**(University of Bristol):

*Random Walks on Vertex-Transitive Graph with Moderate Growth*

**Mikel Eguzki Garciarena Perez**(University of Salerno):

*Automorphisms of regular rooted trees: the Grigorchuck-Gupta-Sidki groups*

**Jorge Fariña Asategui**(Lund University):

*Hausdorff Dimension of Groups Acting on Regular Rooted Trees*

**Axel Péneau**(Université de Rennes 1):

*The universal solenoid*

**Mingkun Liu**(IMJ-PRG):

*Random multi-geodesics on hyperbolic surfaces*

**Polyxeni Spilioti**(Aarhus University):

*On the spectrum of twisted Laplacians and the Teichmüller representation*

**Rares Stan**(IMAR (Inst of Math of the Romanian Academy)):

Uniform Weyl's law on degenerating surfaces

**Tim Höpfner**(Georg-August Universität Göttingen):

Heat Decay on Non-Compact Manifolds

**Ivan Yakovlev**(LaBRI, Université de Bordeaux):

Random square-tiled surfaces

**Carmine Monetta**(University of Salerno):

Problems on graphs associated with a finite group

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