Content
Seminar in Probability Theory
The Seminar in Probability Theory takes place during the semester, normally on Wednesday at 11:00.
Program HS 2018
Date/Time  Speaker  Title  Location 

6 September 2018 
Lisa Hartung New York University 
The Ginibre ensemble and Gaussian multiplicative
chaos >
It was proven by Rider and Virag that the logarithm of the characteristic
polynomial of the Ginibre ensemble converges to a logarithmically correlated random
field. In this talk we will see how this connection can be established on the level
if powers of the characteristic polynomial by proving convergence to Gaussian
multiplicative chaos. We consider the range of powers in the \(L^2\) phase.
(Joint work in progress with Paul Bourgade and Guillaume Dubach). 
Spiegelgasse 1 Room 00.003 
19 September 2018 
Alexander
Drewitz Universität Köln 
Ubiquity of phases in some percolation models with
longrange correlations >
We consider two fundamental percolation models with longrange correlations: The
Gaussian free field and (the vacant set) of Random Interlacements. Both models have
been the subject of intensive research during the last years and decades, on
\(\mathbb Z^d\) as well as on some more general graphs. We investigate some
structural percolative properties around their critical parameters, in particular
the ubiquity of the infinite components of complementary phases.
This talk is based on joint works with A. Prévost (Köln) and P.F. Rodriguez (BuressurYvette). 
Spiegelgasse 1 Room 00.003 
31 October 2018 
Anton Klimovsky Universität DuisburgEssen 
Highdimensional Gaussian fields with isotropic
increments seen through spin glasses >
Finding the (spaceheight) distribution of the (local) extrema of highdimensional
strongly correlated random fields is a notorious hard problem with many
applications. Following Fyodorov and Sommers (2007), we focus on the Gaussian
fields with isotropic increments and take the viewpoint of statistical physics. By
exploiting various probabilistic symmetries, we rigorously derive the
FyodorovSommers formula for the logpartition function in the highdimensional
limit. The formula suggests a rich picture for the distribution of the local
extrema akin to the celebrated spherical SherringtonKirkpatrick model with mixed
pspin interactions.

Spiegelgasse 1 Room 00.003 
7 November 2018 
Dominik
Schröder IST Austria 
Cusp Universality for Wignertype Random Matrices
>
For Wignertype matrices, i.e. Hermitian random matrices with independent, not
necessarily identically distributed entries above the diagonal, we show that at any
cusp singularity of the limiting eigenvalue distribution the local eigenvalue
statistics are universal and form a Pearcey process. Since the density of states
typically exhibits only square root or cubic root cusp singularities, our work
complements previous results on the bulk and edge universality and it thus
completes the resolution of the WignerDysonMehta universality conjecture for the
last remaining universality type.

Spiegelgasse 1 Room 00.003 
14 November 2018 
Marius Schmidt Universität Basel 
Oriented first passage percolation on the
hypercube >
Consider the hypercube as a graph with vertex set \({0,1}^N\) and edges between two
vertices if they are only one coordinate flip apart. Choosing independent standard
exponentially distributed lengths for all edges and asking how long the shortest
directed paths from \((0,..,0)\) to \((1,..,1)\) is defines oriented first passage
percolation on the hypercube. We will discuss the conceptual steps needed to answer
this question to the precision of extremal process following the two paper series
"Oriented first passage percolation in the mean field limit" by Nicola Kistler,
Adrien Schertzer and Marius A. Schmidt: arXiv:1804.03117 [math.PR] and
arXiv:1808.04598 [math.PR].

Spiegelgasse 1 Room 00.003 
21 November 2018 
Antti Knowles University of Geneva 
Local law and eigenvector delocalization for
supercritical ErdosRenyi graphs >
We consider the adjacency matrix of the ErdosRenyi graph \(G(N,p)\) in the
supercritical regime \(pN > C \log N\) for some universal constant C. We show that
the eigenvalue density is with high probability well approximated by the semicircle
law on all spectral scales larger than the typical eigenvalue spacing. We also show
that all eigenvectors are completely delocalized with high probability. Both
results are optimal in the sense that they are known to be false for \(pN < \log N\).
A key ingredient of the proof is a new family of large deviation estimates for
multilinear forms of sparse vectors. Joint work with Yukun He and Matteo Marcozzi.

Spiegelgasse 1 Room 00.003 
28 November 2018 
Gaultier
Lambert University of Zurich 
How much can the eigenvalue of a random matrix
fluctuate? >
The goal of this talk is to explain how much the eigenvalues of large Hermitian
random matrices deviate from certain deterministic locations. These are known as
“rigidity estimates” in the literature and they play a crucial role in the proof of
universality. I will review some of the current results on eigenvalues’
fluctuations and present a new approach which relies on the theory of Gaussian
Multiplicative Chaos and leads to optimal rigidity estimates for the Gaussian
Unitary Ensemble. I will also mention how it is also deduce a central limit theorem
from our proof.
This is joint work with Tom Claeys, Benjamin Fahs and Christian
Webb.

Spiegelgasse 1 Room 00.003 
12 December 2018 
Ioan
Manulescu University of Fribourg 
Uniform Lipschitz functions on the triangular lattice
have logarithmic variations >
Uniform integervalued Lipschitz functions on a finite domain of the triangular
lattice are shown to have variations of logarithmic order in the radius of the
domain. The level lines of such functions form a loop \(O(2)\) model on the edges of
the hexagonal lattice with edgeweight one. An infinitevolume Gibbs measure for
the loop \(O(2)\) model is constructed as a thermodynamic limit and is shown to be
unique. It contains only finite loops and has properties indicative of
scaleinvariance: macroscopic loops appearing at every scale. The existence of the
infinitevolume measure carries over to height functions pinned at 0; the
uniqueness of the Gibbs measure does not. The proof is based on a representation of
the loop \(O(2)\) model via a pair of spin configurations that are shown to satisfy the
FKG inequality. We prove RSWtype estimates for a certain connectivity notion in
the aforementioned spin model.
Based on joint work with Alexander Glazman.

Spiegelgasse 1 Room 00.003 