# Oberseminar Wahrscheinlichkeitstheorie und andere Vorträge im Wintersemester 2020/21

**Organisers**: Noam Berger (TUM), Nina Gantert (TUM), Markus Heydenreich(LMU) Sabine Jansen (LMU), Franz Merkl (LMU), Konstantinos Panagiotou (LMU), Silke Rolles (TUM),

**Talks**:

**Monday**,16^{th} November 2020, 16:00, (using zoom)

Dominik Schmid (TUM)

Title: The TASEP on trees

Abstract: We study the totally asymmetric simple exclusion process (TASEP) on rooted trees. This means that particles are generated at the root and can only jump in the direction away from the root under the exclusion constraint. Our interests are two-fold. On the one hand, we study invariant measures for the TASEP on trees and provide sufficient conditions for the existence of non-trivial equilibrium distributions. On the other hand, we consider the evolution of the TASEP on trees when all sites are initially empty and study currents.This talk is based on joint work with Nina Gantert and Nicos Georgiou.

**Monday**, 23^{th} November 2020, 16:00, (using zoom)

Tyler Helmuth (Universität Durham)

Title: Efficient algorithms for low-temperature spin systems

Abstract: Two fundamental algorithmic tasks associated to discrete statistical mechanics models are *approximate counting* and *approximate sampling*. When correlations are weak (i.e., at sufficiently high temperatures) efficient algorithms for these tasks can be obtained by giving rigorous mixing time bounds for Markov chains. However, at low temperatures, when correlations are strong, mixing times can become impractically large, and Markov chain methods may fail to be efficient. Recently, the cluster expansion has been used to develop provably efficient low-temperature algorithms for some discrete statistical mechanics models. I’ll explain these algorithmic tasks, and how cluster expansion algorithms work, focusing on the particular case of the q-state Potts model on random regular graphs when q is large. If time permits, I’ll also discuss some new probabilistic results that were obtained as consequences of the development of these algorithms.Based on joint work with Matthew Jenssen and Will Perkins.

**Monday**,30^{th} November 2020, 16:00, (using zoom)

Quan Shi (Universität Mannheim)

Title: Diffusions on a space of interval partitions

Abstract: In this talk, we study a family of self-similar interval-partition-valued diffusions with Poisson--Dirichlet pseudo-stationary distribution. Such diffusions arise as limits of certain up-down ordered Chinese restaurant processes and have applications to continuum random tree models. This talk is based on joint works (in progress) with Noah Forman, Douglas Rizzolo, and Matthias Winkel.

**Monday**, 7^{th} December 2020, 16:00, (using zoom)

Pieter Trapman (Universität Stockholm)

Title: Herd immunity, population structure and the second wave of an epidemic

Abstract: The classical herd-immunity level is defined as the fraction of a population that has to be immune to an infectious disease in order for a large outbreak of the disease to be impossible, assuming (often implicitly) that the immunized people are a uniform subset of the population.I will discuss the impact on herd immunity if the immunity is obtained through an outbreak of an infectious disease in a heterogeneous population. The leading example is a stochastic model for two successive SIR (Susceptible, Infectious, Recovered) epidemic outbreaks or waves in the same population structured by a random network. Individuals infected during the first outbreak are (partially) immune for the second one. The first outbreak is analysed through a bond percolation model, while the second wave is approximated by a three-type branching process in which the types of individuals depend on their position in the percolation clusters used for the first outbreak. This branching process approximation enables us to calculate a threshold parameter and the probability the second outbreak is large. This work is based on joined work with Tom Britton and Frank Ball and on ongoing work with Frank Ball, Abid Ali Lashari and David Sirl.

**Monday**, 14^{th} December 2020,

No talk, we refer to Uniqueness methods in statistical mechanics: recent developments and algorithmic applications

**Monday**, 21^{th} December 2020, 16:00, (using zoom)

Finja Ehlers (LMU)

Title: Percolation theory on epidemic models including long distance connections

Abstract: Consider a finite plane square lattice, where the vertices represent the positions of individuals in a population that is exposed to some infectious disease. Assume that the disease can spread the following

way: An infected individual can either be in contact with one of its nearest neighbours and infect it with some probability, or travel to a randomly chosen individual on the lattice and infect it while the

nearest neighbour stays healthy. This talk will start with a short simulation of this model, followed by its mathematical analysis. Travel activities are an important factor for an epidemic, especially if

long distances are involved. For increasing square lattice sizes, the probability of randomly choosing a distant vertex that has been infected before at some fixed time is decreasing. In this case it would

be expected that whenever a distant neighbour is infected, a new cluster is created. Based on this idea, a second similar model is presented, where the distant individual is always infected on a new empty

copy of the lattice instead and the spread can continue on it as well. It will be proved with the coupling method that this model is stochastically dominating the original model.

This work was supervised by Prof. Dr. Franz Merkl.

**Monday**, 11^{th} January 2021, 16:00, (using zoom)

Balint Virag (Universitiy of Toronto)

Title: The heat and the landscape

Abstract: If lengths 1 and 2 are assigned randomly to each edge in the planar grid, what are the fluctuations of distances between far away points?

This problem is open, yet we know, in great detail, what to expect.

The directed landscape, a universal random plane geometry, provides the answer to such questions.

In some models, such as directed polymers, the stochastic heat equation, or the KPZ equation, random plane geometry hides in the background.

Principal component analysis, a fundamental statistical method, comes to the rescue: BBP statistics can be used to show that these models converge to the directed landscape.

**Monday**, 18^{th} January 2021, 16:00, (using zoom)

Siamak Taati (American Universitiy of Beirut)

Title: Positive-rate probabilistic cellular automata with Bernoulli

invariant measures

Abstract: I will sketch the proof of the following result: If a positive-rate

probabilistic cellular automaton has a Bernoulli invariant measure,

then it is ergodic, meaning that it has no other invariant measure and

furthermore it converges (exponentially fast) to its unique invariant

measure. The same is true for (continuous-time) interacting particle

systems. The proof is via the entropy method, but unlike the usual

entropy arguments, does not require the starting measure to be

shift-invariant. I will briefly discuss a practical implication of

this result concerning the (in)feasibility of strictly reversible

computer structures. This is a joint work with Irène Marcovici.

**Monday**, 25^{th} January 2021, 16:00, (using zoom)

Ofer Busani (Universitiy of Bristol)

Title: Non-existence of bi-infinite polymer Gibbs measures on Z^2

Abstract: To each vertex x\in Z^2 assign a positive weight \omega_x. A geodesic between two ordered points on the lattice is an up-right path maximizing the cumulative weight along itself. A bi-infinite geodesic is an infinite path taking up-right steps on the lattice and such that for every two points on the path, its restriction to between the points is a geodesic. Assume the weights across the lattice are i.i.d., does there exist a bi-infinite geodesic with some positive probability? In the case the weights are Exponentially distributed, we answer this question in the negative. We show an analogous result for the positive-temperature variant of this model. Joint work with Marton Balazs and Timo Seppalainen.

**Monday**, 1^{th} February 2021, 16:00, (using zoom)

Lorenzo Taggi (Sapienza Università di Roma)

Title: Exponential decay of correlations for O(N) spin systems for arbitrary N

Abstract: The Spin O(N) model is a classical statistical mechanics model whose configurations are collections of unit vectors, called spins, taking values on the surface of a N -1 dimensional unit sphere, with each spin associated to the vertex of a graph. Some special cases of the spin O(N) model are the Ising model (N = 1), the XY model (N = 2), and the classical Heisenberg model (N = 3). Despite the fact that it is a very classical model, there remain important gaps in understanding, particularly in the case N > 2. This talk will present a new recent result about exponential decay of correlations for arbitrary (non-zero) values of the external magnetic field and arbitrary spin dimension N>1, extending previous results which are only valid for N=1,2,3. Our proof is probabilistic and employs a new representation of the model as a system of “coloured" random walks which is of independent interest.

**Monday**, 8^{th} February 2021, 16:00, (using zoom)

Matthew Dickson (LMU)

Title: Large Deviations of a Spatial Cycle Huang-Yang-Luttinger Loop Soup

Abstract: It is conjectured that the emergence of Bose-Einstein condensation in interacting Bose gases should correspond with the emergence of "infinitely long" cycles in an interacting loop soup. The Huang-Yang-Luttinger (HYL) interaction in the Bose gas is an approximation of the hard-sphere interaction, and here we will relate it to a similar looking interaction on the loop soup. Using large deviation techniques we will derive various important properties of this interacting loop soup and relate them to known properties of the HYL-interacting Bose gas. In particular, we will derive a discontinuous "condensation density" for the loop soup and find a regime in which the large deviation rate function aquires distinct simultaneous minimisers.

**Monday**, 26^{th} April 2021, 16:00, (using zoom)

Gidi Amir (Bar-Ilan University)

Title: TBA

How to get to Garching-Hochbrück