# Oberseminar Wahrscheinlichkeitstheorie und andere Vorträge im Sommersemester 2020

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

**Talks**:

**Monday**, 20^{th} April, 2020, 16:00 (using zoom)

Silke Rolles (TUM)

Title: Recent results on vertex-reinforced jump processes

Abstract: Vertex-reinforced jump processes are stochastic processes

in continuous time that prefer to jump to sites that have

accumlated a large local time. Sabot and Tarrès showed

interesting connections between vertex-reinforced jump

processes and a supersymmetric hyperbolic nonlinear sigma

model introduced by Zirnbauer in a completely different

context.

In the talk, I will present an extension of Zirnbauer's model

and show how it arises naturally as a weak joint limit of a

time-changed version of the vertex-reinforced jump process.

It describes the asymptotics of rescaled crossing numbers,

rescaled fluctuations of local times, asymptotic local times

on a logarithmic scale, endpoints of paths, and last exit trees.

The talk is based on joint work with Franz Merkl and Pierre Tarrès.

**Montag**, 11^{th} May 2020, 16:00, (using zoom)

Mark Peletier(TU Eindhoven, Niederlande)

Title: Continuum limit of a hard-sphere particle system by large deviations

Abstract: Many stochastic particle systems have well-defined continuum limits: as the number of particles tends to infinity, the density of particles converges to a deterministic limit that satisfies a partial differential equation. In this talk I will discuss one example of this. The particle system consists of particles that have finite size: in two and three dimensions they are spheres, in one dimension rods. The particles can not overlap each other, leading to a strong interaction with neighbouring particles. Such systems of particles have been much studied, but for the continuum limit in dimensions two and up there is currently no rigorous result. There are conjectures about the form of the limit equation, often in the form of Wasserstein gradient flows, but to date there are no proofs. We also can not give a proof of convergence in higher dimensions, but in the one-dimensional situation we can give a complete picture, including both the convergence and the gradient-flow structure that derives from the large-deviation behaviour of the particles. This gradient-flow structure shows clearly the role of the free energy and the Wasserstein-metric dissipation, and how they derive from the underlying stochastic particle system. The proof is based on a special mapping of the particle system to a system of independent particles, that is unique to the one-dimensional setup. This mapping is an isometry for the Wasserstein metric, leading to a beautiful connection between limit equations for interacting and non-interacting particle systems. This is joint work with Nir Gavish and Pierre Nyquist.

**Montag**, 18^{th} May 2020, 16:00, (using zoom)

Jeff Steif (University of Gothenburg, Sweden)

Title: Divide and color representations for threshold Gaussian and stable vectors

Abstract: We consider the following simple model: one starts with a set V,

a random partition of V and a parameter p in [0,1]. We then obtain a {0,1}-valued

process indexed by V obtained by independently, for each partition element

in the random partition chosen, with probability p assigning all the elements of the partition

element the value 1, and with probability 1−p, assigning all the elements of the partition

element the value 0. Many models fall into this context: in particular the 0 external field Ising model

(where this is called the Fortuin-Kasteleyn representation).

I will first describe earlier work with Johan Tykesson and then move on to describe

work with Malin Palö Forsström, where we study the question of which threshold Gaussian

and stable vectors have such a representation: (A threshold Gaussian (stable) vector is a vector

obtained by taking a Gaussian (stable) vector and a threshold h and looking where

the vector exceeds the threshold h). The answer turns out to be quite varied depending

on properties of the vector and the threshold; it turns out that h=0 behaves quite

differently than h different from 0. Among other results, in the large h regime, we obtain a

phase transition in the stability exponent alpha for stable vectors where the critical value turns out

to be alpha=1/2.

**Montag**, 25^{th} May 2020, 16:00, (using zoom)

Andrej Nikonov (LMU)

Title: Random limits of random walk induced random graphs

Abstract: Motivated by many characteristics of real world networks such as clustering and power law degree distributions many random graph models reproducing these have been introduced. Processes shaping real world networks are often also local, i.e. they often rely on properties of the network in the neighbourhood of a vertex. A random walk can be regarded as such a local selection process for creating or reinforcing edges. In the talk we look at a process where repetitively a n-step random walk from a random starting vertex A to vertex B leads to the reinforcement of the edge from A to B. Different approaches to analyse this process and in particular associated random limits are discussed.

**Monday, **8^{th}June 2020, 16:00, (using zoom)

Marek Biskup (UCLA)

Title: TBA

**Monday, **15^{th}June 2020, 16:00, (using zoom)

Franziska Kühn (TU Dresden)

Title: Regularity theory for non-local operators

Abstract: Let $A$ be the infinitesimal generator of a Lévy process. Classical examples are, for instance, the Laplacian (generator of Brownian motion) and the fractional Laplacian (generator of isotropic stable Lévy process). In this talk, we study the regularity of solutions $f$ to the Poisson equation $Af=g$. We show how gradient estimates for the transition density of the Lévy process can be used to obtain Hölder estimates for $f$. Moreover, we present a Liouville theorem for Lévy operators: If $f$ is a solution to $Af=0$ which is at most of (suitable) polynomial growth, then $f$ is a polynomial. We illustrate our results with examples and discuss some possible generalizations.

**Monday, **22^{th}June 2020, 16:30, LMU, room B252, Theresienstr. 39, Munich

Sabine Jansen (LMU)

Title: Phase transitions for a hierarchical mixture of cubes

Abstract: We consider a discrete toy model for phase transitions in mixtures of incompressible droplets. The model consists of non-overlapping hypercubes in Z^d with side-lengths 2^j, j\in N_0. Cubes belong to an admissible set B such that if two cubes overlap, then one is contained in the other, a picture reminiscent of Mandelbrot's fractal percolation model. I will present exact formulas for the entropy, discuss phase transitions from a fluid phase with small cubes towards a condensed phase with a macroscopic cube, and explain how the toy model fits into a renormalization program for mixtures of hard spheres in R^d. Based on arXiv:1909.09546 (J. Stat. Phys. 179 (2020), 309-340).

**Monday, **29^{th}June 2020, 16:00, (using zoom)

Sébastien Ott (Università degli Studi Roma Tre)

Title: TBA

**Montag**, 6^{th} July 2020, 16:00, (Virtuelle Veranstaltung)

Noam Berger (TUM)

Title:TBA

**Montag**, 13^{th} July 2020, 16:00, (Virtuelle Veranstaltung)

Dirk-André Deckert (LMU)

Title:TBA

**Montag**, 20^{th} July 2020, 16:00, (Virtuelle Veranstaltung)

David Criens (TUM)

Title:TBA

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