# Workshop Women in Probability 2018

The scientific program is organized by Noam Berger, Diana Conache, Nina Gantert, Silke Rolles and Sabine Jansen (LMU). 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 Silvia Schulz.

## Location and dates

**Friday afternoon, 8 ^{th} June**, and

**Saturday morning, 9**, at Zentrum Mathematik, Technische Universität München.

^{th}June 2018All talks take place in room FMI 00.07.014 on the ground floor of the mathematics building. Zentrum Mathematik is situated in Boltzmannstr. 3 in Garching near Munich. How to go there?

## Speakers

- Fabienne Castell (Universität Marseille)
- Karen Habermann (Universität Bonn)
- Irene Marcovici (Université de Lorraine)
- Sara Merino-Aceituno (University of Sussex)
- Francesca Nardi (University of Florence)
- Ellen Powell (ETH Zürich)
- Ellen Saada (CNRS-Université Paris Descartes)
- Kristina Schubert (Universität Münster)

## Program

**Friday, June 8 ^{th}, 2018**:

- 14:00-14:45
**Ellen Saada** - 15:00-15:45
**Francesca Nardi** - 15:45-16:15 Coffee Break
- 16:15-17:00
**Sara Merino Aceituna** - 17:15-18:00
**Kristina Schubert**

We will go for dinner after the talks.

**Saturday, June 9 ^{th}, 2018**:

- 09:00-09:45
**Fabienne Castel** - 09:55-10:40
**Irene Marcovici** - 10:40-11:00 Coffee Break
- 11:00-11:45
**Ellen Powell** - 11:55-12:40
**Karen Habermann**

## Titles and abstracts

**Fabienne Castell**: Multiresolution analysis on graphs using random forests and Markov process intertwining

Abstract: Multiscale analysis of discretized functions defined on a regular grid, such as audio signals, is a classical problem in signal processing. Fast and efficient algorithms using wavelet transforms have been proposed to handle this problem. However, many data are defined on finite connected weighted graphs which are not regular grids, raising the need of algorithms designed for such irregular structures. We propose a new method for performing multiscale analysis of functions defined on the vertices of a finite graph. Our approach relies on probabilistic tools: a random spanning forest to downsample the set of vertices, and approximate solutions of Markov intertwining relation to provide a subgraph structure and a filterbank which is a basis of the set of functions.**Karen Habermann**: On sub-Riemannian diffusion bridges

Abstract: We discuss small-time phenomena for sub-Riemannian diffusion bridges. These are stochastic processes which live in a constrained geometry, a so-called sub-Riemannian geometry, and which are conditioned by their initial and final positions. By providing a specific example, we show that sub- Riemannian diffusion bridges can exhibit exotic behaviours, i.e. qualitatively different behaviours compared to Brownian bridges. We further study the small-time fluctuations for sub-Riemannian diffusion loops and prove the weak convergence of suitably rescaled fluctuations to a limiting diffusion loop, which we identify explicitly in terms of a certain local limit operator.**Irene Marcovici**: Ergodicity of some classes of cellular automata subject to noise

Abstract: Cellular automata (CA) are dynamical systems on symbolic configurations on the lattice. They are also used as models of massively parallel computers. As dynamical systems, one would like to understand the effect of small random perturbations on the dynamics of CA. As models of computation, they can be used to study the reliability of computation against noise. We consider various families of CA and prove that they are highly unstable against noise, meaning that they forget their initial conditions under slightest positive noise. This is manifested as the ergodicity of the resulting probabilistic CA. For instance, we prove that if the deterministic model is either nilpotent (highly stable) or permutive (highly chaotic), its small random perturbations are ergodic. We are also interested in the sensitivity of tilings against noise, and I will briefly discuss the links between these two problems.**Sara Merino-Aceituno**: Stochastic particle systems mean-field limit in wave turbulence

Abstract: The isotropic 4-wave kinetic equation appears in the theory of wave turbulence. We consider its weak formulation using model homogeneous kernels. We study the existence and uniqueness of solutions and the mean-field limit from stochastic particle systems.**Francesca Nardi**: Tunneling behavior of Ising and Potts model in the low-temperature regime

Abstract: We consider the ferromagnetic q-state Potts model with zero external field in a finite volume and assume that the stochastic evolution of this system is described by a Glauber-type dynamics parametrized by the inverse temperature . Our analysis concerns the low-temperature regime, in which this multi-spin system has q stable equilibria, corresponding to the configurations where all spins are equal. Focusing on grid graphs with various boundary conditions, we study the tunneling phenomena of the q-state Potts model. More specifically, we describe the asymptotic behavior of the first hitting times between stable equilibria in probability, in expectation, and in distribution and obtain tight bounds on the mixing time as side-result.In the special case q = 2, our results characterize the tunneling behavior of the Ising model on grid graphs. This is a joint work with A. Zocca.**Ellen Powell**: Characterising the Gaussian Free Field.

Abstract: The planar Gaussian free field is the natural generalisation of Brownian motion to two-dimensional time, and over the past few years has been conjectured and/or proved to arise as a universal scaling limit from a broad range of statistical physics models. I will give an overview of these results, and discuss a recent work with Nathanaël Berestycki and Gourab Ray, showing that the GFF is characterised by conformal invariance and a certain domain Markov property.**Ellen Saada**: zero-range process in random environment

Abstract: In this talk I will consider a zero-range process with site disorder. This one-dimensional, nearest-neighbor, attractive dynamics with a bounded jump rate, exhibits a phase transition: there are no invariant measures above some critical density. In collaboration with C. Bahadoran, T. Mountford and K. Ravishankar, we have first obtained necessary and sufficient conditions for weak convergence to the critical invariant measure. We have then derived the hydrodynamical behavior of the system, and finally, we have proven local equilibrium results, and a dynamical loss of mass.**Kristina Schubert**: Spacings in Random Matrix Theory

Abstract: We consider random matrices from unitary invariant matrix ensembles and more general repulsive particle systems. The considered eigenvalue statistic is the spacing distribution, i.e. the empirical distribution of nearest neighbor spacings. We review some classical results for spacings and establish the convergence of the spacing distribution in a rather strong sense to a universal limiting distribution if the matrix size tends to infinity. Here, we use a non-linear rescaling, called unfolding, to transform the ensemble such that the eigenvalue density is asymptotically constant. The main ingredient for the proof is a strong bulk universality result for correlation functions in the unfolded setting.