# Workshop Women in Probability 2017

The scientific program is organized by Noam Berger, Diana Conache, Nina Gantert, and Silke Rolles. This conference is supported by the "Women for Math Science Program" at Technische Universität München.

There is no conference fee and there are no gender restrictions on the audience, everybody is welcome to attend.

For hotel reservations, please contact Wilma Ghamam.

## Location and dates

**Friday afternoon, 21 ^{th} July**, and

**Saturday morning, 22**, at Zentrum Mathematik, Technische Universität München.

^{th}July 2017All talks take place in room FMI 00.07.011 on the ground floor of the mathematics building. Zentrum Mathematik is situated in Boltzmannstr. 3 in Garching near Munich. How to go there?

## Speakers

- Codina Cotar (University College London)
- Aurelia Deshayes (Université Paris Diderot)
- Margherita Disertori(Universität Bonn)
- Mylene Maida (Université des Sciences et Technologies de Lille)
- Constanza Rojas-Molinas (Rheinische Friedrich-Wilhelms-Universität Bonn )
- Kavita Ramanan (Brown University)
- Wioletta Ruszel (Technical University of Delft)
- Anita Winter (Universität Duisburg-Essen)

## Program

**Friday, July 21th, 2017**:

- 14:00-14:45
**Codina Cotar**: Density functional theory and many-marginals optimal transport with Coulomb and Riesz costs - 15:00-15:45
**Mylene Maida**: Concentration of measure for Coulomb gase*s* - 15:45-16:15 Coffee Break
- 16:15-17:00
**Constanza Rojas-Molinas**: The transport exponent in disordered quantum systems - 17:15-18:00
**Wioletta Ruszel**: Odometers and bi-Laplacian fields

We will go for dinner after the talks.

**Saturday, July 22 ^{th}, 2017**:

- 09:00-09:45
**Margherita Disertori**: Some results on history dependent stochastic processes: the case of vertex reinforced jump process - 09:55-10:40
**Aurelia Deshayes**: Scaling limit of subcritical contact process - 10:40-11:00 Coffee Break
- 11:00-11:45
**Kavita Ramanan**: Tales of Random Projections: where probability theory meets geometry - 11:55-12:40
**Anita Winter**: Real trees versus algebraic trees as state of stochastic processes

## Titles and abstracts

**Codina Cotar**: Density functional theory and many-marginals optimal transport with Coulomb and Riesz costs

Abstract: Multi-marginal optimal transport with Coulomb cost arises as a dilute limit of density functional theory, which is a widely used electronic structure model. The number N of marginals corresponds to the number of particles. I will discuss the question whether “Kantorovich minimizers” must be “Monge minimizers” (yes for N = 2, open for N > 2, no for N =infinity), and derive the surprising phenomenon that the extreme correlations of the minimizers turn into independence in the large N limit. I will also discuss the next order term limit and the connection of the problem to Coulomb and Riesz gases. The talk is based on joint works with Gero Friesecke (TUM), Claudia Klueppelberg (TUM), Brendan Pass (Alberta) and Mircea Petrache.**Mylene Maida**: Concentration of measure for Coulomb gases Abstract : A Coulomb gas is the canonical Gibbs measure associated with a system of particles in electrostatic interaction. As the number of particles grows to infinity, the empirical measure of a Coulomb gas converges weakly towards an equilibrium measure, characterized by a variational principle. We obtain sub-gaussian concentration inequalities around this equilibrium measure, in the weak and Wasserstein topologies. This yields for instance a concentration inequality at the correct rate for the Ginibre ensemble. The proof relies on new functional inequalities, which are counterparts of Talagrand's transport inequality in the Coulomb interaction setting. Joint work with Djalil Chafaï and Adrien Hardy.**Constanza Rojas-Molinas**: The transport exponent in disordered quantum systems Abstract: In this talk we will study some dynamical properties of the Anderson model, a Schrödinger operator with random potential used to study the (absence of) electron propagation in disordered quantum systems. We focus on the dynamical transport exponent that measures the propagation of wave packets and its relation with the decay of Green's function. Namely, we generalize a result by F. Germinet and A. Klein for the one-particle Anderson model concerning the relation between transport exponent and the applicability of the multiscale analysis method used in the proof of localization. This gives a characterization of the regions of localization and delocalization in the spectrum. We then generalize these results to the N-particle Anderson model and to random unitary operators. Based on joint work with A. Klein and S. Nguyen, and on joint work with O. Bourget.**Wioletta Ruszel**: Odometers and bi-Laplacian fields Abstract: The divisible sandpile model is a special case of the class of continuous fixed energy sandpile models on some lattice or graph where the initial configuration is random and the evolution deterministic. One question which arises is under which conditions the model will stabilize or not. The amount of mass u(x) emitted from a certain vertex x during stabilization is called the odometer function. In this talk we will construct the scaling limit of the odometer function of a divisible sandpile model on a torus and show that it it converges to a continuum bi-Laplacian field. This is joint work with A. Cipriani (U Bath) and R. Hazra (ISI Kolkatta).**Margherita Disertori**: Some results on history dependent stochastic processes: the case of vertex reinforced jump process. Abstract: Vertex reinforced jump process was introduced by Werner in 2000 as an an example of history dependent process, where the particle tends to come back more often to sites it has already visited in the past. I will give an overview on the model and explain some recent results.**Aurelia Deshayes**: Scaling limit of subcritical contact process

Abstract: I will talk about subcritical contact process on Zd. The contact process, introduced in 1974 by Harris, models the spread of an infection. It is one of the simplest interacting particle systems which exhibits a phase transition. In the subcritical case, the process vanishes if we start with a finite number of infected particles. But what happens if we start with infinite number of particles? I will present a work, in collaboration with Leo Rolla, where we study the subcritical contact process for large times starting with all sites infected. The configuration is described in terms of the macroscopic locations of infected regions in space and the relative positions of infected sites in each such region (which involce a quasi stationary distribution of the contact process modulo translation). This work is an extension of a previous paper written by Andjel, Ezanno, Groisman and Rolla which describes the subcritical contact process seen from the rightmost infected particle in dimension 1.-
**Kavita Ramanan**: Tales of Random Projections: where probability theory meets geometry

Abstract: The structure of high-dimensional measures is a fascinating subject whose study leads to an interesting interplay between geometry and probability. Classical theorems in probability theory such as the central limit theorem and Cramer's theorem can be viewed as providing information about certain scalar projections of high-dimensional product measures. This talk will focus on the behavior of random projections of more general (possibly non-product) high-dimensional measures, which are of interest in diverse fields, ranging from asymptotic convex geometry to high-dimensional statistics. Although the study of (typical) projections of high-dimensional measures dates back to Borel, only recently has a theory begun to emerge, which in particular identifies the role of certain geometric assumptions that lead to better behaved projections. In this talk we will describe the tail behavior of random projections of general (non-product) high-dimensional measures, which are of interest in diverse fields, ranging from asymptotic convex geometry to high-dimensional statistics. Along the way, we will explain why Cramer's theorem is atypical and describe large deviation results on the Stiefel manifold. Our results serve to complement the central limit theorem for convex sets, and its extensions. This talk is based on various joint works with Steven Kim and Nina Gantert. **Anita Winter**: Real trees versus algebraic trees as state of stochastic processes

Abstract: In this talk we are interested in continuum trees as limit objects of graph-theoretic trees when the number of vertices goes to infinity. Depending on which notion of convergence we choose, different objects are obtained. A notion of convergence with many applications at different places is based on encoding trees as metric measure spaces and then using the Gromov-weak topology. Apparently such an encoding is not suitable in all applications. Problems arise in the construction of scaling limits of Markov chains taking values in graph-theoretic trees whenever the metric and the measure have a slightly different scaling behavior. We therefore introduce algebraic trees which have a nice encoding as triangulations of the circle and which can be seen as metric measure trees where we have forgotten the metric. We will explain the two approaches and illustrate them with the example of Aldous chain on cladograms.