# Workshop Women in Probability 2011

### July 15-16, 2011 at Zentrum Mathematik, TU München

The scientific program, which is organized by Nina Gantert and Silke Rolles features talks by:

- Hanna Döring (Berlin)
- Uta Freiberg (Siegen)
- Erika Hausenblas (Leoben)
- Sandra Kliem (Eindhoven)
- Noemi Kurt (Berlin)
- Anne Fey (Amsterdam).
- Eva Löcherbach (Cergy-Pontoise)
- Sylvie Roelly (Potsdam)

All talks take place in room MI 00.07.011 of the mathematics building. It is located on the ground floor. Zentrum Mathematik is situated in Boltzmannstr. 3 in Garching near Munich. How to go there?

Note that there are no gender restrictions on the audience.

This conference is supported by the ``Women for Math Science Program'' at Technische University München.

## Program

**Friday, July 15**- 14:00-14:45 Uta Freiberg: Spectral analysis of random Sierpinski gaskets
- 15:00-15:45 Hanna Döring: Moderate deviations for the eigenvalue counting function of Wigner matrices
- 16:15-17:00 Noemi Kurt: A random interface subject to attractive and repulsive constraints
- 17:15-18:00 Anne Fey: Limiting shapes for nonabelian sandpile growth models
- We will go for dinner after the talks.

**Saturday, July 16**- 9:00-9:45 Sylvie Roelly: (Infinite) systems of Brownian balls and their equilibrium states
- 9:55-10:40 Erika Hausenblas: The stochastic Reaction Diffusion Equation driven by Levy noise
- 11:00-11:45 Sandra Kliem: Renormalisation of Hierarchically Interacting Lambda-Cannings Processes
- 11:55-12:40 Eva Löcherbach: Hitting time moments and speed of convergence to equilibrium
- We will go for lunch after the talks.

## Titles and abstracts

- Hanna Döring:
*Moderate deviations for the eigenvalue counting function of Wigner matrices*

**Abstract:**In the joint work with Peter Eichelsbacher, we establish a moderate deviation principle for the number of eigenvalues of a Wigner matrix in an interval. The proof relies on fine asymptotics of the variance of the eigenvalue counting function of matrices in the Gaussian Unitary Ensemble due to Gustavsson. The extension to large families of Wigner matrices is based on the Tao and Vu Four Moment Theorem and applies localization results by Erdös, Yau and Yin. - Uta Freiberg:
*Spectral analysis of random Sierpinski gaskets*

**Abstract:**Self similar fractals are often used in modeling porous materials. However, the assumption of strict self similarity could be too restricting. So, we present several models of random fractals which could be used instead. After recalling the classical approaches of random homogenous and recursive random fractals, we show how to interpolate between these two models with the help of so called V-variable fractals. This concept (developed by Barnsley, Hutchinson & Stenflo) allows the definition of new families of random fractals, hereby the parameter V describes the degree of "variability" of the realizations.We discuss how the degree of variability influences the geometric, analytic and stochastic behaviour of these sets. - Erika Hausenblas:
*The stochastic Reaction Diffusion Equation driven by Levy noise*

**Abstract:**Within the talk I will present my joint work with Brzezniak about the existence of the Stochastic Reaction Diffusion Equation with Levy noise. In the first part I will present the preliminaries, i.e. stochastic integration in Banach spaces and maximal Inequalities. In the second part, I will outline the existence of a solution due to a SPDE with only continuous coefficients and will explain the proof. Finally I will apply this result to get the existence of the solution to a stochastic Reaction Diffusion Equation driven by Levy noise. - Sandra Kliem:
*Renormalisation of Hierarchically Interacting Lambda-Cannings Processes*

**Abstract:**We consider interacting populations on the hierarchical group of order N. The dynamics of the interacting populations involve migration between sites of blocks of varying sizes k and resampling inside blocks of varying sizes k. Migration is modeled via a random walk kernel on each block level. The resampling is modeled via so-called interacting Lambda-Cannings processes that arise as the continuum-mass limits of Cannings models and are dual to Lambda-coalescents. In one reproductive step an ancestor is chosen from the current population of the block and creates a positive fraction of children in the block.

The large-scale space-time behaviour is investigated by a renormalisation analysis of block-averages on successive space-time scales combined with a hierarchical mean-field limit. Results on regions for clustering and coexistence of types are given.

This is joint work with Andreas Greven, Frank den Hollander and Anton Klimovsky. - Noemi Kurt:
*A random interface subject to attractive and repulsive constraints*

**Abstract:**Consider a lattice-based model for random interfaces or membranes, given as the Gaussian field with the discrete biharmonic Green's function as covariance matrix. The purpose of this choice of covariance matrix is to incorporate curvature of the interface into the model. We want to understand the behaviour of this interface subject to certain constraints. In this talk, we consider two cases: On one hand a pinning force which gives a reward if the interface stays close to a hyperplane, on the other hand a repulsive force that pushes the interface away from a certain region of space. The analysis of the model requires a detailed understanding of the properties of discrete biharmonic Green's functions with various boundary conditions, for which a representation in terms of intersections of random walk is helpful. In sufficiently high dimensions, we obtain good bounds on the covariances, which enable us to prove positivity of the free energy in the pinning case, and the precise height of repulsion in the second case. This is joint work with Erwin Bolthausen and Alessandra Cipriani (Zürich). - Anne Fey:
*Limiting shapes for nonabelian sandpile growth models*

**Abstract:**On a grid, start with a pile of height n > 1 at the origin, and a pile of height h < 1 on every other grid vertex. Split every pile that has height at least 1, that is, distribute its total height equally among the neighbors. We are interested in the set S of sites where we split a pile at least once. Depending on h, n and possibly even on the order in which we split piles, either S reaches a final size or S keeps increasing. We show that the first case occurs for h small enough and arbitrary splitting order, and we give bounds for the size of S, using similar methods as for the abelian sandpile growth model. In the second case, our model behaves strikingly different from the abelian sandpile growth model. With the parallel splitting order, S can have a multitude of different limiting shapes. We demonstrate several of them, leaving plenty of opportunity for further research.

Joint work with Haiyan Liu. - Eva Löcherbach:
*Hitting time moments and speed of convergence to equilibrium*

**Abstract:**We establish equivalences between existence of certain hitting time moments for recurrent Markov processes and their speed of convergence to equilibrium. We will consider this question both from a probabilistic point of view (where we are in particular interested in non asymptotic deviation inequalities) and from an analytical point of view (functional inequalities). This talk will focus mostly on the case where the rate of convergence is polynomial. It is based on a joint work with Dasha Loukianova (Evry) and Oleg Loukianov (Fontainebleau). - Sylvie Roelly:
*(Infinite) systems of Brownian balls and their equilibrium states*

**Abstract:**We consider an infinite system of non overlapping globules undergoing Brownian motions in R3. The term globules means that the objects we are dealing with are spherical, but with a radius which is random and time-dependent. The dynamics is modelized by an infinite-dimensional Stochastic Differential Equation with local time. Existence and uniqueness of a strong solution is proven for such an equation. We also describe a class of equilibrium measures.