Seminar on Stochastic Processes

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Autumn Semester 2022

Date / Time

Speaker

Title

Location

28 September 2022
17:15-18:15

Prof. Dr. Robin Pemantle
University of Pennsylvania

Event Details

Seminar on Stochastic Processes

Title	Concepts of negative dependence for binary random variables
Speaker, Affiliation	Prof. Dr. Robin Pemantle, University of Pennsylvania
Date, Time	28 September 2022, 17:15-18:15
Location	HG G 19.1
Abstract	This talk reviews 50-60 years of the theory of negative dependence of binary random variables, beginning with origins in mathematical statistics and statistical mechanics. This culminates in the Borcea-Branden-Liggett theory, which connects negative dependence to the geometry of zero sets of polynomials. It provides a reasonably checkable condition, which is satisfied in many examples, and has strong consequences such as negative association. The last part of the talk focuses on more recent (in the last ten years) work of various people. This concerns concentration inequalities, Lorentzian measures, and a CLT based on the geometry of zeros. The talk will end with some open problems.

Concepts of negative dependence for binary random variablesread_more

HG G 19.1

12 October 2022
17:15-18:15

Dr. Piet Lammers
IHES / Université Paris-Saclay

Event Details

Seminar on Stochastic Processes

Title	Planarity, percolation, and height functions
Speaker, Affiliation	Dr. Piet Lammers, IHES / Université Paris-Saclay
Date, Time	12 October 2022, 17:15-18:15
Location	HG G 19.1
Abstract	Fröhlich and Spencer proved the Berezinskii-Kosterlitz-Thouless transition in 1981, through a relation with delocalisation of height functions. Their delocalisation proof goes through a relation with the Coulomb gas. In recent years it is becoming clear that this phase transition can also be understood in a simpler way through couplings with planar percolation models. This talk presents one such delocalisation proof and outlines some other recent advancements.

Planarity, percolation, and height functionsread_more

HG G 19.1

19 October 2022
17:15-18:15

Dr. Paul Thevenin
Uppsala University

Event Details

Seminar on Stochastic Processes

Title	Fragmentation of trees and drifted excursions
Speaker, Affiliation	Dr. Paul Thevenin, Uppsala University
Date, Time	19 October 2022, 17:15-18:15
Location	HG G 19.1
Abstract	The fragmentation of a tree is a process which consists in cutting the tree at random points, thus splitting it into smaller connected components as time passes. In the case of the so-called Brownian tree, it turns out that the sizes of these subtrees, known as the Aldous-Pitman fragmentation process, have the same distribution as the lengths of the excursions over its current infimum of a linearly drifted Brownian excursion, as proved by Bertoin. We provide a natural coupling between these two objects. To this end, we make use of the so-called cut-tree of the Brownian tree, which can be seen as the genealogical tree of the fragmentation process. Joint work with Igor Kortchemski.

Fragmentation of trees and drifted excursionsread_more

HG G 19.1

26 October 2022
17:15-18:15

Alice Contat
Mathématiques Orsay

Event Details

Seminar on Stochastic Processes

Title	Parking on the infinite binary tree
Speaker, Affiliation	Alice Contat, Mathématiques Orsay
Date, Time	26 October 2022, 17:15-18:15
Location	HG G 19.1
Abstract	Consider a rooted tree whose vertices will be interpreted as free parking spots, each spot accommodating at most one car. On top of that tree, we consider a non-negative integer labeling representing the number of cars arriving on each vertex. Each car tries to park on its arrival vertex, and if the spot is occupied, it travels downwards in direction of the root of the tree until it finds an empty vertex to park. If there is no such vertex on the path towards the root, the car exits the tree, contributing to the flux of cars at the root. This models undergoes an interesting phase transition which we will analyze in detail. After an overview of the case where the underlying tree is a critical Bienaymé—Galton—Watson tree, we will concentrate on the case where the underlying tree is the infinite binary tree, where the phase transition which turns out to be “discontinuous”. The talk is based on a joint work with David Aldous, Nicolas Curien and Olivier Hénard.

Parking on the infinite binary treeread_more

HG G 19.1

2 November 2022
17:15-18:15

Prof. Dr. Antti Knowles
Université de Genève

Event Details

Seminar on Stochastic Processes

Title	Spectral phases of Erdös-Rényi graphs
Speaker, Affiliation	Prof. Dr. Antti Knowles, Université de Genève
Date, Time	2 November 2022, 17:15-18:15
Location	HG G 19.1
Abstract	Disordered quantum systems exhibit a variety of spectral phases, characterized by the extent of spatial localization of the eigenvectors. Through their adjacency matrices, random graphs provide a natural class of models for such systems, where the disorder arises from the random geometry of the graph. The simplest random graph is the Erdös-Rényi graph G(N,p), whose adjacency matrix is the archetypal sparse random matrix. The parameter d=pN represents the expected degree of a vertex. A dramatic change in behaviour is known to occur at the scale d \sim \log N, which is the threshold where the degrees of the vertices cease to concentrate. Below this scale the graph becomes inhomogeneous and develops structures such as hubs and leaves which accompany the appearance of a localized phase. I report on recent progress in establishing the phase diagram for G(N,p) at and below the critical scale d \sim \log N. We show that the spectrum splits into a fully delocalized region in the middle of the spectrum and a semilocalized phase near the spectral edges. The transition between the phases is sharp in the sense of a discontinuity in the localization exponent of eigenvectors. Furthermore, we show that the semilocalized phase consists of a fully localized region and in addition, for some values of d, a complementary region that we conjecture to be nonergodic delocalized. Joint work with Johannes Alt and Raphael Ducatez.

Spectral phases of Erdös-Rényi graphsread_more

HG G 19.1

23 November 2022
17:15-18:15

Prof. Dr. Francisco Caravenna
Università degli Studi di Milano-Bicocca

Event Details

Seminar on Stochastic Processes

Title	The critical 2d Stochastic Heat Flow
Speaker, Affiliation	Prof. Dr. Francisco Caravenna, Università degli Studi di Milano-Bicocca
Date, Time	23 November 2022, 17:15-18:15
Location	HG G 19.1
Abstract	We consider the 2-dimensional Stochastic Heat Equation (SHE), which falls outside the scope of existing solution theories for singular stochastic PDEs. When we regularise the SHE by discretising space-time, the solution can be identified with the partition function of a statistical mechanics model, the so-called directed polymer in random environment. We prove that as the discretisation is removed and the noise strength is rescaled in a critical way, the solution converges to a unique continuum limit: a universal process of random measures on R^2, which we call the critical 2d Stochastic Heat Flow. We investigate its features, showing in particular that it cannot be the exponential of a generalised Gaussian field. Based on joint work with R. Sun and N. Zygouras.

The critical 2d Stochastic Heat Flowread_more (CANCELLED)

HG G 19.1

30 November 2022
17:15-18:15

Diederik van Engelenburg
Institut für Mathematik, Universität Wien

Event Details

Seminar on Stochastic Processes

Title	The number of ends in the uniform spanning tree
Speaker, Affiliation	Diederik van Engelenburg, Institut für Mathematik, Universität Wien
Date, Time	30 November 2022, 17:15-18:15
Location	HG G 19.1
Abstract	I will talk about a result with Tom Hutchcroft, stating that the number of ends of the uniform spanning tree (UST) is almost surely equal to the number of ends of the underlying graph in the context of recurrent stationary random rooted graphs. Together with previous results in the transient case, this completely resolves the problem of the number of ends of wired uniform spanning forest components in stationary random rooted graphs and confirms a conjecture of Aldous and Lyons (2006). Our work elaborates on work with Nathanael Berestycki, in which we relate the end-structure of the UST on recurrent graphs to potential theoretic properties of the underlying graphs.

The number of ends in the uniform spanning treeread_more

HG G 19.1

7 December 2022
17:15-18:15

Dr. Alexis Prévost
Université de Genève

Event Details

Seminar on Stochastic Processes

Title	Generating Galton-Watson trees using random walks and percolation for the Gaussian free field
Speaker, Affiliation	Dr. Alexis Prévost, Université de Genève
Date, Time	7 December 2022, 17:15-18:15
Location	HG G 19.1
Abstract	Consider the percolation problem induced by the level sets of the Gaussian free field on a supercritical Galton-Watson tree. It was proved by Abächerli and Sznitman that the associated critical parameter is positive as long as the mean offspring distribution m is at least two. We extend this result to all supercritical Galton-Watson trees, that is m>1. I will explain why the positivity of the critical parameter is typically harder to obtain in the regime 1

Generating Galton-Watson trees using random walks and percolation for the Gaussian free fieldread_more

HG G 19.1

14 December 2022
17:15-18:15

Prof. Dr. Christophe Garban
Université Lyon 1

Event Details

Seminar on Stochastic Processes

Title	Debye screening, hyperuniformity and GFF fluctuations for the Coulomb gas on Z^d
Speaker, Affiliation	Prof. Dr. Christophe Garban, Université Lyon 1
Date, Time	14 December 2022, 17:15-18:15
Location	HG G 19.1
Abstract	The goal of this talk will be to present some puzzling properties of the (two-component) lattice Coulomb gas on the d-dimensional lattice. The connection of this model with integer-valued fields and compact-valued spin systems will be emphasised through the talk. This is a joint work with Avelio Sepúlveda.

Debye screening, hyperuniformity and GFF fluctuations for the Coulomb gas on Z^dread_more

HG G 19.1

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