# Colloquium in probability and other talks in summer term 2019

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

**Talks**:

**Monday**, 6^{th} May 2019, 16:30, TUM, room 2.01.10, Parkring 11, Garching-Hochbrück (Technische Universität München)

Clement Cosco (Université Paris Diderot)

Title: Approxmation of the Kardar-Parisi-Zhang equation in Weak Disorder in d ≥ 3 (joint work with Francis Comets and Chiranjib Mukherjee)

Abstract: Trying to give a proper definition of the KPZ (Kardar-Parisi-Zhang) equation in dimension d ≥ 3 is a challenging question. A plan to do so, is to first consider the well-defined KPZ equation when the white noise is smoothed in space. Then, the solution of the smoothed equation can be expressed as the free energy of a continuous polymer. Using this representation for d ≥ 3 and small noise intensity, it is possible to study the asymptotic behavior of the solutions in the limit where the smoothing is removed. In particular, we can prove that the stationary solution is a good approximation of the solutions for a wide range of initial conditions. We can further give quantitative estimates of this approximation when the initial profile is flat, which will involve the Gaussian free field in R^d.

**Monday**, 13^{th} May 2019, 16:30, TUM, room 2.01.10, Parkring 11, Garching-Hochbrück (Technische Universität München)

Martin Slowik (Technische Universität Berlin)

Title: From stochastic interface models to random walks in time-dependent random environments -- and back?

Abstract: There is a somewhat unexpected connection between random walks in time-dependent random environments and gradient Gibbs measures describing stochastic interfaces in systems arising from statistical mechanics, e.g., the Ginzburg-Landau model, and its dynamics. After reviewing how the space-time covariances of the height of the interface can be expressed in terms of random walks among dynamic random conductances, I will discuss recent progress on the understanding of the behaviour of such random walks. A particular emphasis will be on the results and the methods that has been used to prove invariance principles and local limit theorems for almost every realisation of the environment.

**Monday**, 20^{th} May 2019, 16:30, LMU, room B252, Theresienstr. 39, Munich

Diana Conache (TUM)

Title: Dislocation lines in three-dimensional solids at low temperature

Abstract: We propose a model for three-dimensional solids on a mesoscopic scale with a statistical mechanical description of dislocation lines in thermal equilibrium. The model has a linearized rotational symmetry, which is broken by boundary conditions. We show that this symmetry is spontaneously broken in the thermodynamic limit at small positive temperatures. In particular, we will focus on the statistical mechanical properties of a random Burgers vector configuration, in the spirit of the Fröhlich-Spencer approach to the Villain model. We prove via a cluster expansion a Gaussian lower bound for the Fourier transform of an observable with respect to the thermal measure. This is joint work with Roland Bauerschmidt, Markus Heydenreich, Franz Merkl and Silke Rolles.

**Monday**, 27^{th} May 2019, 16:30, TUM, room 2.01.10, Parkring 11, Garching-Hochbrück (Technische Universität München)

Maximilian Nitzschner (ETH Zürich)

Title: Disconnection by Gaussian Free Field level sets and entropic repulsion

Abstract: We consider level set percolation for the Gaussian Free Field (GFF) on the Euclidean lattice in dimensions larger or equal to three, in a strongly percolative regime. We study the 'disconnection event' that the level set below a given level disconnects the discrete blow-up of a compact set A from the boundary of an enclosing box. In particular, we give asymptotic large deviation upper and lower bounds on the probability of disconnection that extend earlier results by Sznitman. Moreover, we study the behaviour of local averages of the GFF conditionally on disconnection: If certain critical levels coincide, we show that the GFF experiences a push down to a level that is locally governed by a multiple of the harmonic potential of A, which may be seen as an instance of entropic repulsion. This is a joint work with Alberto Chiarini.

**Monday**, 3^{rd }June 2019, 16:30, TUM, room 2.01.10, Parkring 11, Garching-Hochbrück (Technische Universität München)

Manfred Denker (Penn State University)

Title: Local Limit Theorems And Applications

Abstract: In the talk I shall discuss a lattice local limit theorem for Gibbs-Markov processes and sketch applications to the Poincare exponent of some Fuchsian group and local times for continued fractions and fractal Gaussian noise.

**Thursday**, 13^{th} June 2019, 16:30, TUM, room 2.01.10, Parkring 11, Garching-Hochbrück (Technische Universität München)

Evita Nestoridi (Cambridge)

Title: Mixing time of the upper triangular matrix walk over Z/mZ.

Abstract: We study a natural random walk over the upper triangular matrices, with entries in Z/mZ, generated by steps which add or subtract row $i + 1$ to row $i$. We show that the mixing time of the lazy random walk is $O(n^2m^2)$ which is optimal up to constants. This generalizes a result of Peres and Sly and answers a question of Stong and of Arias-Castro, Diaconis and Stanley. This is joint work with Allan Sly.

**Monday**, 17^{th} June 2019, 16:30, LMU, room B252, Theresienstr. 39, Munich

Gabriel Oexle (MSc presentation)

Title: Cluster expansion for Poisson-saddlepoint approximation

Abstract: The intensity of a Gibbs point process is an intractable function in the model parameters. Therefore, approximations such as the Poisson-saddlepoint approximation are needed. With the help of cluster expansion, we show that the Poisson-saddlepoint approximation converges to the true intensity if the dimension of the underlying metric space goes to infinity. Cluster expansion is a method originally established in mathematical physics. If the interactions are weak, the hope is that the Gibbs point process is close to the non-interacting Poisson point process and we may hopefully capture the correction terms by a convergent power series in the activity parameter z. The correction terms of the intensity depend on a sum over all connected graphs. The correction terms of the Poisson-saddlepoint approximation, on the other hand, depend only on the sum over all trees. We show that as the dimension goes to infinity, the contributions of the connected graphs, which are not trees, vanishes.

**Monday**, 1^{th} July 2019, 16:30, LMU, room B252, Theresienstr. 39, Munich

Sandra Kliem (Universität Frankfurt)

Title: The one-dimensional contact process and the KPP equation with noise

Abstract: The one-dimensional KPP-equation driven by space-time white noise, % \[ \partial_t u = \partial_{xx} u + \theta u - u^2 + u^{\frac{1}{2}} dW, \qquad t>0, x \in \mathbb{R}, \theta>0, \qquad \qquad u(0,x) = u_0(x) \geq 0 \] % is a stochastic partial differential equation (SPDE) that exhibits a phase transition for initial non-negative finite-mass conditions. Solutions to this SPDE arise for instance as (weak) limits of approximate densities of occupied sites in rescaled one-dimensional long range contact processes. If $\theta$ is below a critical value $\theta_c$, solutions with initial finite mass die out to $0$ in finite time, almost surely. Above this critical value, the probability of (global) survival is strictly positive. Let $\theta>\theta_c$, then there exist stochastic wavelike solutions which travel with non-negative linear speed. For initial conditions that are ‘’uniformly distributed in space’’, the corresponding solutions are all in the domain of attraction of a unique non-zero stationary distribution. For the (parameter-dependent) nearest-neighbor contact process on $\mathbb{\ZZ}$, an interacting partic le system, more is known. A complete convergence theorem holds, that is, a full description of the limiting law of a solution is available, starting from any initial condition. Its proof relies in essence on the progression of so-called edge processes. In these models, edge speeds characterize critical values. In my talk, I will introduce the two models in question (nearest-neighbor contact process and KPP-equation with noise). Then I explain in how far the concepts and techniques of the first model can be used to obtain new insights into the second model. In particular, the problems one encounters when changing from the discrete to the continuous (in space) setting are highlighted and approaches to resolve them are discussed.

**Monday**, 8^{th} July 2019, 16:30,TUM, room 2.01.10, Parkring 11, Garching-Hochbrück (Technische Universität München)

Christoph Thäle (Ruhr-Universität Bochum)

Title:How to cut an apple, or splitting tessellations in spherical spaces

Abstract: The concept of splitting tessellation processes in spherical spaces is introduced using Markov process theory. Expectations, variances and covariances of spherical curvature measures induced by a splitting tessellation are studied using tools from spherical integral geometry. Also the spherical pair-correlation function of the surface measure is computed explicitly.

**Im Anschluss daran** (ca. 17:30):

Joscha Prochno (Universität Graz)

Title: The asymptotic volume ratio of the Schatten classes

Abstract: The unit ball of the finite-dimensional Schatten trace class $S_p^n$ consists of all real n x n matrices A whose vector of singular values belongs to B_p^n, where p>0. Saint Raymond [Studia Math. 80, 63–75, 1984] proved an asymptotic formula for the volume of the Schatten unit ball including a non-specific constant for which he provided both lower and upper bounds. We determine the precise limiting constant based on ideas from the theory of logarithmic potentials with external fields. As an application we compute the precise asymptotic volume ratio of the Schatten p-balls, as n → ∞, thereby extending Saint Raymond’s estimate in the case of the nuclear norm (p = 1) to the full regime 1 ≤ p ≤ ∞ with exact limiting behavior. (joint work with Zakhar Kabluchko and Christoph Thäle)

**Monday**, 15^{th} July 2019, 16:30, TUM, room 2.01.10, Parkring 11, Garching-Hochbrück (Technische Universität München)

Simon Gabriel (TUM)

Title: Condensation in the Reinforced Branching Process

Abstract: Dereich, Mailler and Mörters (2017) recently analysed a multi-type branching process with reinforcement in continuous time with type-space [0,1]. An individual's reproduction depends on its type, the closer to one the higher its rate of reproduction. Under certain conditions a finite fraction of the limiting population will consist of individuals having maximal fitness; a phenomenon which is known as condensation. We will give an intuitive introduction into the model and present a necessary and sufficient condition, namely non-existence of a Malthusian parameter, for condensation to occur.

**Monday**, 22^{th} July 2019, 16:30, LMU, room B252, Theresienstr. 39, Munich

Peter Gracar (Universität zu Köln)

Title: Spread of infection by random walks - Multi-scale percolation along a Lipschitz surface

Abstract: A conductance graph on $\mathbb{Z}^d$ is a nearest-neighbor graph where all of the edges have positive weights assigned to them. We first consider a point process of particles on the nearest neighbour graph $(\mathbb{Z}^d,E)$ and show some known results about the spread of infection between particles performing continuous time simple random walks. Next, we extend consider the case of uniformly elliptic random graphs on $\mathbb{Z}^d$ and show that the infection spreads with positive speed also in this more general case. We show this by developing a general multi-scale percolation argument using a two-sided Lipschitz surface that can also be used to answer other questions of this nature. Joint work with Alexandre Stauffer.

**Monday**, 26^{th} August 2019, 16:30, LMU, room B252, Theresienstr. 39, Munich

Sam Thomas (Universität Cambridge)

Title: Cutoff for Random Walk on Random Cayley Graphs

Abstract: Consider the random Cayley graph of a finite group $G$ with respect to $k$ generators chosen uniformly at random, with $1 \ll \log k \ll \log|G|$: the vertices are the group elements, and $g,h \in G$ are connected if there exists a generator $z$ so that $g = hz$ or $gz = h$. A conjecture of Aldous and Diaconis asserts that the simple random walk on this graph exhibits cutoff, at a time which depends only on $|G|$ and $k$, not on the algebraic structure of the group $G$ (ie universally in $G$). We verify this conjecture for a wide class of Abelian groups. Time permitting, we discuss extensions to the case where the underlying group $G$ is non-Abelian. There the cutoff time cannot be written only as a function of $|G|$ and of $k$; it depends on the algebraic structure. This is joint work with Jonathan Hermon

How to get to Garching-Hochbrück