# Mini-Workshop "Women in Probability" 2013

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.

Childcare can be provided during the workshop. If you would like to use it please inform Wilma Ghamam.

# Location and dates

**Friday, 5th July**, and **Saturday, 6th July 2013**, at Zentrum Mathematik, Technische Universität München.

All 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

- Alessandra Bianchi (Università degli Studi di Padova, Italy)
- Alessandra Faggionato (Università degli Studi di Roma "La Sapienza", Italy)
- Sabine Jansen (Universiteit Leiden, The Netherlands)
- Malwina Luczak (Queen Mary, University of London, United Kingdom)
- Wioletta Ruszel (Delft University of Technology, The Netherlands)
- Ellen Saada (CNRS, Université Paris Descartes, France)
- Amanda Turner (Lancaster University, United Kingdom)
- Anita Winter (Universität Duisburg-Essen, Germany)

# Program

Friday, 5th July 2013:

- 14:30-15:15 Amanda Turner: The emergence of branching in Hastings-Levitov type random clusters
- 15:30-16:15 Malwina Luczak: A fixed-point approximation for a routing in equilibrium
- 16:45-17:30 Alessandra Faggionato: Large deviations of empirical flow and current
- 17:45-18:30 Alessandra Bianchi: Metastability in Markovian systems via quasi-stationary measures and capacities

We will go for dinner after the talks.

Saturday, 6th July 2013:

- 9:00-9:45 Ellen Saada: Couplings and attractiveness for interancting particle systems
- 9:55-10:40 Sabine Jansen: Metastability at low temperature for continuum interacting particle systems
- 11:00-11:45 Wioletta Ruszel: Cascades on trees
- 11:55-12:40 Anita Winter: Convergence of bi-measure R-trees and the subtree prune process

We will go for lunch after the talks.

# Titles and abstracts

**Alessandra Bianchi**: Metastability in Markovian systems via quasi-stationary measures and capacities

Abstract: In this talk I will present some recent results, obtained in collaboration with A. Gaudilliere, concerning the characterization of metastability for Markovian systems on finite configuration spaces. The two main objects entering this characterization are quasi-stationary measures and capacities. After recalling their definition (and useful generalizations), I will discuss their connection with metastable systems and review the main results, including sharp estimates on mean exit time and transition time, and the proof of their asymptotic exponential laws. Some examples and applications of the method will be finally discussed.

**Alessandra Faggionato**: Large deviations of the empirical flow and current

Abstract: We describe a joint large deviation principle for the empirical measure and flow for a continuous time Markov chain. By projection, we recover information on the large deviations of the empirical current and show a Gallavotti-Cohen symmetry for it. Finally, we discuss concrete examples and applications.

**Sabine Jansen**: Metastability at low temperature for continuum interacting particle systems

Abstract: We consider a system of point particles in a finite box in**R**^{2}that interact via a finite-range attractive pair potential, and move according to a Markov process that has the grand-canonical Gibbs measure as a reversible measure. The chemical potential is such that the system favors a packed box, but has a nucleation barrier to overcome in order to go from an empty box to a filled box. We are interested in the nucleation time in the limit as the temperature tends to zero. We use the potential-theoretic approach to metastability. The results should extend earlier work for lattice systems; the main difficulty lies in understanding the energy landscape of the continuum particle system, a problem of intrinsic interest in analysis.

This talk reports on joint work in progress with Frank den Hollander.

**Malwina Luczak**: A fixed-point approximation for a routing model in equilibrium

Abstract: The following dynamic route-allocation model was studied fairly extensively in the 1990s. There are a number of nodes, and calls arrive between each pair as a Poisson process. Where possible, a call is allocated to the direct link between the two nodes. However, the link between each pair of nodes has a fixed capacity: when the direct link is at capacity, some number of two-link paths is inspected, and one of them is used for the incoming call if possible. The duration of each call is an exponential random variable with fixed mean. It is natural to expect that, in equilibrium, around each node, the number of links of each load will be an approximation to the fixed point of a certain differential equation, provided there is just one fixed point. We show that this is indeed true provided the arrival rate is sufficiently small or sufficiently large. We also survey what is known about this model.

This is joint work with Graham Brightwell.

**Wioletta Ruszel**: Cascades on trees

Abstract: Inspired by experiments in neuroscience studying self-organized critical behaviour in the brain we study the abelian sandpile model on a random trees. It was proven by Dhar and Majumdar (1990) that for the full binary tree (and Bethe lattice) the probability that an avalanche is of size*k*decays as a power-law with mean-field exponent^{3}/_{2}. For the binary and binomial tree, using a transfer matrix approach introduced by Dhar & Majumdar, we prove exponential decay of correlations, and in a small supercritical region (i.e., where the branching process survives with positive probability) exponential decay of avalanche sizes. This shows a phase transition phenomenon between exponential decay and power law decay of avalanche sizes. Finally we discuss some extensions and work in progress to avalanche size statistics of sandpile models on Galton-Watson trees.

This is joint work with Antal Jarai, Fank Redig and Ellen Saada.

**Ellen Saada**: Couplings and attractiveness for interacting particle systems

Abstract: Attractiveness for particle systems corresponds to the existence of a coupling of two processes with the same infinitesimal generator, that stay ordered as soon as it is the case for their initial states. I will mainly focus on generalized misanthrope models. They are conservative particle systems on**Z**^{d}for which the "basic coupling" construction is not possible under necessary and sufficient conditions for attractiveness. For such models, in each transition, k particles may jump from a site x to another site y, with k≥1, and the jump rate depends on the number of particles only at sites x and y. Under attractiveness conditions, I will explain the increasing coupling we have constructed, and how it permits to determine the extremal invariant and translation invariant measures for the dynamics. I will present examples of generalized zero-range, generalized target and generalized misanthrope models. Finally I will explain how to deal with attractiveness for exclusion processes with speed change, and for non-conservative dynamics.

This is joint work with Thierry Gobron (CNRS, Cergy-Pontoise) and Lucie Fajfrova (UTIA, Prague).

**Amanda Turner**: The emergence of branching in Hastings-Levitov type random clusters

Abstract: In 1998 Hastings and Levitov proposed a one-parameter family of models for planar random growth in which clusters are represented ascompositions of conformal mappings. This family includes physically occurring processes such as diffusion-limited aggregation (DLA), dielectric breakdown and the Eden model for biological cell growth. In the simplest case of the model (corresponding to the parameter α=0), James Norris and I showed how the Brownian web arises in the limit resulting from small particle size and rapid aggregation. In particular this implies that beyond a certain time, all newly aggregating particles share a single common ancestor. I shall show how small changes in alpha result in the emergence of branching structures within the model so that the beyond a certain time, the number of common ancestors is a random number whose distribution can be obtained.

This is based on joint work with Fredrik Johansson Viklund (Columbia) and Alan Sola (Cambridge).

**Anita Winter**: Convergence of bi-measure R-trees and the subtree prune process

Abstract: In 1998 Aldous and Pitman constructed a tree-valued Markov chain by pruning off more and more subtrees above randomly chosen edges of a Galton-Watson tree. More recently Abraham, Delmas and He considered a similar process, where the cut-points are chosen in a degree-dependent way. In the same spirit prunings of continuum trees were studied by various authors. However, so far no precise link between the prunings of discrete and continuum trees has been given. In this talk we encode trees as metric measure spaces and equip them additionally with a pruning measure, and provide a topology on the space of bi-measure R-trees. We then construct THE subtree prune process and show that convergence of initial states implies convergence of the whole bi-measure valued paths.

This is joint work with Wolfgang Löhr and Guillaume Voisin.