# Workshop Women in Probability 2019

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, 31 ^{th} May**, and

**Saturday morning, 1**, at Zentrum Mathematik, Technische Universität München.

^{th}June 2019All 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

- Luisa Andreis (Weierstraß-Institut)
- Gioia Carinci (Delft University of Technology)
- Hanna Döring (Universität Osnabrück)
- Lisa Hartung (Johannes Gutenberg-Universität Mainz)
- Cecile Mailler (University of Bath)
- Eveliina Peltola (University of Geneva)
- Elena Pulvirenti (Universität Bonn)
- Ecaterina Sava-Huss (Graz University of Technology)

## Program

**Friday, May 31 ^{th}, 2019**:

- 14:00-14:45 Luisa Andreis:
- 15:00-15:45 Eveliina Peltola:
**15:45-16:15 Coffee Break**- 16:15-17:00 Lisa Hartung: The Ginibre characteristic polynomial and Gaussian Multiplicative Chaos
- 17:15-18:00 Ecaterina Sava-Huss
**:**Aggregation models based on random walks and rotor walks

We will go for dinner after the talks.

**Saturday, June ^{1th}, 2019**:

- 09:00-09:45 Cecile Mailler: The monkey walk: A random walk with random reinforced relocations and fading memory
- 09:55-10:40 Gioia Carinci
**:**Inclusion process, sticky brownian motion and condensation - 10:40-11:00
**Coffee Break** - 11:00-11:45 Hanna Doering:
- 11:55-12:40 Elena Pulvirenti: The Widom-Rowlinson model: metastability, mesoscopic and microscopic fluctuations for the critical droplet

## Titles and abstracts

**Luisa Andreis:**A large-deviations approach to the multiplicative coagulation process

Abstract: At least since the days of Smoluchovski, there is a desire to understand the behaviour of large particle systems that undergo chemical reactions of coagulation type. One of the phenomena that attracts much attention is the question for the existence of a phase transition of gelation type, i.e., the appearance of a particle of macroscopic size in the system. In this talk, we consider the (non-spatial) coagulating model (sometimes called the Marcus-Lushnikov model), starting with N particles with mass one each, where each two particles coagulate after independent exponentially distributed times that depend on a given coagulation kernel, function of the two masses. We focus on the case in which the corresponding coagulation kernel is multiplicative in the two masses, hence the process is identied as the multiplicative coagulation process. This case is of particular interest also for its strong relations with the time dependent Erdös-Renyi random graph. We work for xed time t > 0 and derive, for the number N of initial particles going to innity, a joint large-deviations principle for all relevant quantities in the system (microscopic, mesoscopic and macroscopic particle sizes) with an explicit rate function. We deduce laws of large numbers and in particular derive from that the well-known phase transition at time t = 1, the time at which a macroscopic particle (the so-called gel) appears, as well as the Smoluchovski characterisation of the statistics of the nite-sized particles. This is a joint work with Wolfgang Konig and Robert Patterson (WIAS).**Gioia Carinci:**Inclusion process, sticky brownian motion and condensation

Abstract: The inclusion process is an interacting particle system modeling the motion of agents diffusing with an attractive interaction. I will discuss its behavior in a suitable condensation regime where the attractive part of the dynamics is predominant. In particular an explicit formula is obtained for expectation and variance of the fluctuation density field. This result shows, from a microscopic point of view, the formation of a condensed state where particles pile up. The proof is based on duality and the study of the two-particle dual process that is proved to converge to sticky Brownian particles. Joint work with C. Giardina, F. Redig**Hanna Döring:**Limit theorems in dynamic random networks

Abstract: Many real world phenomena can be modelled by dynamic random networks. We will focus on preferential attachment models where the networks grow node by node and edges with the new vertex are added randomly depending on a sublinear function of the degree of the older vertex. Using Stein’s method provides rates of convergence for the total variation distance between the evolving degree distribution and an asymptotic power-law distribution as the number of vertices tends to infinity. This is a joint work with Carina Betken and Marcel Ortgiese.**Lisa Hartung:**The Ginibre characteristic polynomial and Gaussian Multiplicative Chaos

Abstract: 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 whole so-called subcritical phase. (Joint work in progress with Paul Bourgade and Guillaume Dubach).**Cecile Mailler:**The monkey walk: A random walk with random reinforced relocations and fading memory

Abstract: In this joint work with Gerónimo Uribe-Bravo, we prove and extend results from the physics literature about a random walk with random reinforced relocations. The "walker" evolves in $\mathbb Z^d$ or $\mathbb R^d$ according to a Markov process, except at some random jump-times, where it chooses a time uniformly at random in its past, and instatnly jumps to the position it was at that random time. This walk is by definition non-Markovian, since the walker needs to remember all its past. Under moment conditions on the inter-jump-times, and provided that the underlying Markov process verifies a distributional limit theorem, we show a distributional limit theorem for the position of the walker at large time. The proof relies on exploiting the branching structure of this random walk with random relocations; we are able to extend the model further by allowing the memory of the walker to decay with time.**Eveliina Peltola:**Crossing probabilities of multiple Ising interfaces

Abstract: I discuss crossing probabilities of multiple interfaces in the critical Ising model with alternating boundary conditions. In the scaling limit, they are conformally invariant expressions given by so-called pure partition functions of multiple SLE(kappa) with kappa=3. I also describe analogous results for critical percolation and the Gaussian free field. This is joint work with Hao Wu (Yau Center / Tsinghua University).**Elena Pulvirenti:**The Widom-Rowlinson model: metastability, mesoscopic and microscopic fluctuations for the critical droplet

Abstract: In this talk I will discuss 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 the volume of the union of the disks minus the sum of the volumes 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 find that, in the limit of low temperature, the critical droplet is close to a disk of a certain deterministic critical radius, with a boundary that is random and consists of a large number of disks that stick out by a small distance. We compute two terms in the asymptotic scaling of the average crossover time, corresponding to the volume free energy and the surface free energy of the critical droplet, respectively. In order to derive these results we need detailed control of the microscopic and mesoscopic fluctuations of the surface of the critical droplet in the static version of the Widom-Rowlinson model. Our proofs rely on large deviation principles for the volume of the halo, moderate deviation principles for the surface of the halo, and isoperimetric inequalities. This is a joint work in progress with F. den Hollander, S. Jansen, R. Kotecky.**Ecaterina Sava-Huss:**Aggregation models based on random walks and rotor walks

Abstract: