# 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: A quenched invariance principle for random walks with long range jumps

Abstract: I will discuss random walks among random conductances on the hypercubic lattice that allow for jumps of arbitrary length. This includes the random walk on the long-range percolation graph obtained by adding to $\mathbb Z^d$ an edge between $x$ and $y$ with probability proportional to $|x-y|^{-s}$, independently of other pairs of vertices. By a combination of functional inequalities and location-dependent truncations, I will prove that the random walk scales to Brownian motion under a diffusive scaling of space and time. The proof follows the usual route of reducing the statement to everywhere sublinearity of the corrector. We prove the latter under moment conditions on the environment that in fact turn out to be more or less necessary for the method of proof. For the above percolation problem, this requires the exponent~$s$ to exceed~$2d$. Based on joint work with X. Chen, T. Kumagai and J. Wang.

**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, 16:00, (using zoom)

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: Scaling limit of low temperature interfaces in 2D Potts model

Abstract: In this talk, I will discuss various problems linked with 2D interfaces: "free" (Dobrushin) interface, interface above a hard wall, pinning and wetting problems. I will first formulate them in the Potts model and introduce a toolbox to treat them. The main result I will focus on is the construction of a coupling of the interface with a random walk in a potential, using Ornstein-Zernike theory. As application of this coupling, one can derive the scaling limit of the interface in the situations previously mentioned.

Based on joint work with Ioffe, Velenik, Wachtel, and on work of Campanino, Ioffe, Velenik.

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

Noam Berger (TUM)

Title: Stationary Hastings-Levitov process

Abstract: The Hastings-Levitov process, introduced by Hastings and Levitov in 1998, is a planar aggregation process in which at every time a new particle attaches itself to the existing cluster at a point which is determined by the harmonic measure. This model was studied extensively in recent years. The main advantage of this model is that its direct connection to complex analysis makes it tractable. The main disadvantage is some non-physical behaviour of the particle sizes. In this talk I will present a new half-plane variant of the Hastings-Levitov model, and will demonstrate that our variant, called the Stationary Hastings-Levitov, maintains the tractability of the original model, while avoiding the non-physical behavior of the particle sizes.

The talk is based on joint work with Jacob Kagan, Eviatar Procaccia and Amanda Turner.

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

Dirk-André Deckert (LMU)

Title:An introductory survey on supervised learning

Abstract: Machine learning is an area of research that is spread across many disciplines such as computer science, mathematics, and neuroscience. Its sub-field supervised learning recently gave rise to many advances. I will review its main mathematical questions and afterwards give an introduction on the feasibility of learning from a statistical point of view. In this regard, I will review the mathematical PAC framework introduced by Valiant and discuss the characterization of PAC learnability of binary classification in terms of finite Vapnik–Chervonenkis dimension.

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

David Criens (TUM)

Title: A Harnack Inequality for Random Walks in Balanced Environments

Abstract: We consider random walks in balanced i.i.d. non-elliptic random environments (RWBRE). Similar as Brownian motion is related to the heat equation, RWBRE is related to random difference equations. We discuss a parabolic Harnack principle for these equations. The talk is based on joint work with Noam Berger.

**Montag**, 27^{th} July 2020, 16:00, (Virtuelle Veranstaltung)

Kilian Weyers (TUM)

Title: Martingale defocusing and transience of a self-interacting random walk

Abstract: We consider the following martingale dispersion result proved by Peres, Schapira and Sousi: If, up to time n, the jumps of a martingale are bounded from above by log(n)^a (with some positive a < 1) and the conditional variance of each jump is at least 1, then P(M_n = 0) gets arbitrarily small for sufficiently large n. After presenting the main ideas of the proof, we discuss why there is no such dispersion result in the case a=1.

Peres, Schapira and Sousi used a more general version of this dispersion result to show that the following random walk W=(X,Y,Z) in Z^3 is transient: When visiting a vertex for the first time, Z changes by +/-1, while on later visits (X,Y) changes by (+/-1,0) or (0,+/-1). We look at this application as well and present the main ideas of the proof that W is transient.

Everything is based on the following paper: Yuval Peres, Bruno Schapira, and Perla Sousi. Martingale defocusing and transience of a self-interacting random walk. Ann. Inst. H. Poincaré Probab. Statist., 52(3):1009–1022, 2016.

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