# Workshop Women in Probability 2015

The scientific program is organized by Noam Berger, 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, 3 ^{rd} July**, and

**Saturday, 4**, at Zentrum Mathematik, Technische Universität München.

^{th}July 2015All talks take place in room 2.02.01, Parkring 11, Garching-Hochbrück (Technische Universität München). How to go there?

## Speakers

- Oriane Blondel (Université Claude Bernard Lyon 1, France)
- Alessandra Cipriani (Weierstrass Institute, Berlin, Germany)
- Loren Coquille (Universität Bonn, Germany)
- Alessandra Faggionato (Universita di Roma, Italia)
- Veronique Gayrard (Université Aix-Marseille, France)
- Noemi Kurt (Technische Universität Berlin, Germany)
- Francesca Nardi (Eindhoven University of Technology, Netherlands)
- Elena Pulvirenti (Leiden University)

## Program

Friday, July 3, 2015:

- 14:00-14:45 Francesca Nardi: Asymptotic behavior of hitting times for Metropolis Markov chains and applications to the hard-core model on grids
- 15:00-15:45 Alessandra Faggionato: Mott random walk
- 16:15-17:00 Alessandra Cipriani: Extremes of the supercritical Gaussian Free Field

17:15-18:00 Oriane Blondel: Random walk on the East model (and other environments with spectral gap)

We will go for dinner after the talks.

Saturday, July 4, 2015:

- 09:00-09:45 Elena Pulvirenti: Metastability for the Widom-Rowlinson model
- 09:55-10:40 Loren Coquille: On the Gibbs states of the Ising and Potts models
- 11:00-11:45 Noemi Kurt: The seed bank coalescent
- 11:55-12:40 Veronique Gayrard: Aging in mean-field spin-glasses

## Titles and abstracts

**Francesca Nardi:**Asymptotic behavior of hitting times for Metropolis Markov chains and applications to the hard-core model on grids

Abstract: We consider a stochastic model known as hard-core model where particles in a finite volume are subject to hard-core constrains and study the transition time between feasible configurations of this model. Such hard-core constraints arise in various area, such as statistical physics, combinatorics and communication networks. In the hard core model on the two-dimensional grid with periodic boundary conditions there are 2 global minima: the even configuration (respectively odd) where all particles are on the even chessboard (respectively odd). We study the tunneling time between even and odd chessboard and give estimates in probability in law and in distribution. The work I will present is in collaboration with S. Borst, J.S.H. van Leeuwaarden and A. Zocca.**Alessandra Faggionato:**Mott random walk

Abstract: Mott random walk is a random walk in a random environment used to model phonon-assisted variable range hopping conduction in disordered solids. In the talk we will discuss some physical properties, including the Mott-Efros-Shklovskii law for the decay of diffusivity at low temperature, and give a survey of rigorous mathematical results. Finally, we will mention some more recent progresses con Mott random walk under the effect of a homogeneous external field.**Alessandra Cipriani:**Extremes of the supercritical Gaussian Free Field

Abstract: We show that the rescaled maximum of the discrete Gaussian Free Field in dimension larger or equal to $3$ is in the maximal domain of attraction of the Gumbel distribution. We show that the result follows from an interesting application of the Stein-Chen method by Arratia, Goldstein, and Gordon (1989). This is a joint work with A. Chiarini and R. S. Hazra.**Oriane Blondel:**Random walk on the East model (and other environments with spectral gap)

Abstract: The East model is a one-dimensional interacting particle system with non attractive spin-flip dynamics. In the physics literature, it is a key example of a model with glassy features. Here we take this model as a random environment and investigate the behaviour of a random walk whose jump rates depend on the current configuration. The analysis relies on general results established for random walks in random environment when the environment is Markovian with positive spectral gap.**Elena Pulvirenti:**Metastability for the Widom-Rowlinson model

Abstract: In this paper we study the Widom-Rowlinson model on a finite two-dimensional box subject to a stochastic dynamics in which particles are randomly created and annihilated inside the box according to an infinite reservoir with a given chemical potential. The particles are viewed as points carrying disks and the energy of a particle configuration is equal to minus the volume of the total overlap of the disks. Consequently, the interaction between the particles is attractive. We are interested in the metastable behaviour of the system at low temperature when the chemical potential is supercritical. In particular, we start with the empty box and are interested in the first time when the box is fully covered by disks. In order to achieve the transition from empty to full, the system needs to create a sufficiently large droplet, called critical droplet, which triggers the crossover. We compute the distribution of the crossover time, identify the size and the shape of the critical droplet, and investigate how the system behaves on its way from empty to full. This is a joint work in progress with F. den Hollander, S. Jansen, R. Kotecky'.**Loren Coquille****:**On the Gibbs states of the Ising and Potts models

Abstract : I will first review what is known about the set of Gibbs states of the Ising and Potts models, mentioning recent joint works with H. Duminil-Copin, D. Ioffe and Y. Velenik for the 2d case. A natural question (open in general) is to determine whether any Gibbs state is weak limit of finite-volume measures with deterministic boundary conditions. I will give a counter-example in the 3d Ising case, and point out what the issues are in order to extend it to the Potts model. I'll end with a few conjectures.**Noemi Kurt:**The seed bank coalescent

Abstract: Seed banks occur in populations where individuals may enter a reversible stage of inactivity that may be of some duration. In the biological literature it is generally believed that the presence of seed banks leads to an increased genetic variablity, and that seed banks may act as a buffer against other evolutionary forces. In this talk, we introduce a new mathematical model for a population where individuals may take `dormant forms', and identify a new natural coalescent structure, the seed-bank coalescent, which describes the gene genealogy of such populations. The qualitatively new feature of the seed-bank coalescent is that ancestral lineages are independently blocked at a certain rate from taking part in coalescence events, thus strongly altering the predictions of classical coalescent models. We discuss the long-time behaviour of the population model and the corresponding coalescent, and show that even thought fixation of one genetic type happens almost surely, the time to fixation is much longer than in classical population models. In the retrospective picture, we show that, the seed-bank coalescent `does not come down from infinity', and the time to the most recent common ancestor is highly elevated. This provides a genealogical explanation for the increase in genetic variability. (joint work with J. Blath, A. Gonzalez Casanova, M. Wilke Berenguer (all TU Berlin).**Veronique Gayrard:**Aging in mean-field spin-glasses

Abstract: As evidenced by an extensive body of experiments, glassy systems are never in equilibriumon laboratory time-scales; instead, their dynamics become increasingly slower as time elapses.The physically interesting issue is thus that of ageing, a property of time-time correlation functionsthat characterizes the slow decay to equilibrium characteristic for these systems.In recent years, the mathematical analysis of several models (of mean-field spin-glass dynamics,but not only) has revealed a universal mechanism behind this phenomenon: it linksaging to the arcsine law for L´evy subordinators through the asymptotic behavior of a suitabletime change of the dynamics called clock-process. This clock process is a partial sum processof dependent random variables with a random distribution.In this talk I will present a general method for proving functional limit theorems for clockprocesses and explain how to apply it to the REM and the p-spin Sherrington-Kirkpatrickmodel, first for a simple dynamics (the random hopping dynamics) and then for Metropolisdynamics of the REM.