# Workshop Women in Probability 2014

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, 13 ^{th} June**, and

**Saturday, 14**, at Zentrum Mathematik, Technische Universität München.

^{th}June 2014All 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

- Anne-Laure Basdevant (Universite Paris 10, France)
- Margherita Disertori (Universität Bonn, Germany)
- Bénédicte Haas (Université Paris-Dauphine, France)
- Elena Kosygina (The City University of New York, USA)
- Anna Levit (The University of British Columbia, Vancouver, Canada)
- Kavita Ramanan (Brown University, Providence, USA)
- Adela Svejda (Technion - Israel's Institute of Technology, Haifa, Israel)

# Program

Friday, June 13, 2014:

- 14:00-14:45 Kavita Ramanan: Obliquely reflected diffusions in non-smooth domains
- 15:00-15:45 Elena Kosygina: Excursions of excited random walks on integers
- 16:15-17:00 Adela Svejda: Clock processes on infinite graphs and aging in Bouchaud's asymmetric trap model
- 17:15-18:00 Anna Levit: Brownian excursions and critical quantum random graphs

We will go for dinner after the talks.

Saturday, June 14, 2014:

- 9:30-10:15 Bénédicte Haas: Scaling limits of
*k*-ary growing trees - 10:45-11:30 Margherita Disertori: Some results on history dependent stochastic processes
- 11:45-12:30 Anne-Laure Basdevant: The shape of large balls in highly supercritical percolation

# Titles and abstracts

**Anne-Laure Basdevant**: The shape of large balls in highly supercritical percolation

Abstract: In this talk, I will describe a connection between the distances in the infinite percolation cluster on the plane and the discrete-time TASEP on**Z**. This will show that, when the parameter of the percolation model goes to one, large balls in the cluster are asymptotically shaped near the axes like arcs of parabola.

**Margherita Disertori**: Some results on history dependent stochastic processes

Abstract: Edge reinforced random walk (ERRW) and vertex reinforced jump processes are history dependent stochastic processes, where the particle tends to come back more often on sites it has already visited in the past. For a particular scheme of reinforcement these processes are mixtures of reversible Markov chains whose mixing measure can be related to a non-linear sigma model introduced in the context of random matrices. I will give an overview on these models and explain some recent results.

**Bénédicte Haas**: Scaling limits of*k*-ary growing trees

Abstract: For each integer k>1, we introduce a sequence of k-ary discrete trees constructed recursively by choosing at each step an edge uniformly among the present edges and grafting on "its middle" k-1 new edges. When k=2, this corresponds to a well-known algorithm which was first introduced by Rémy to generate random trees uniformly distributed among the set of rooted binary trees with n labeled leaves. Our main result concerns the asymptotic behavior of these trees as the number of steps n becomes large: for all k, the sequence of k-ary trees grows at speed n^{{1/k}}towards a k-ary random real tree that belongs to the family of self-similar fragmentation trees. This convergence is proved with respect to the Gromov-Hausdorff-Prokhorov topology. We also study embeddings of the limiting trees when k varies.

Based on a joint work with Robin Stephenson.

**Elena Kosygina**: Excursions of excited random walks on integers

Abstract: We shall discuss results concerning the depth and duration of excursions of excited random walks (ERWs) in i.i.d. bounded "cookie environments". One of the highlights of the talk is a macroscopic description of a transient ERW conditioned to return to its starting point. Using this description we are able to exhibit a new phase transition for ERWs, namely, between weak and strong transience*. In particular, our results show that for ERWs there is a range of parameters for which the walk is ballistic, i.e. has a non-zero linear speed, but is not strongly transient.

The talk is partially based on a joint work with M. Zerner (Universität Tübingen).

*Transient ERW is said to be strongly transient if the expected time of return to its starting point, conditioned on return, is finite; otherwise it is said to be weakly transient.

**Anna Levit**: Brownian excursions and critical quantum random graphs

Abstract: Let us consider a continuum analog of Erdős-Rényi random graph – the quantum random graph. In this graph the vertices are replaced by circles punctured at Poisson points of arrivals, and connections are derived through another Poisson process on the circle. In this talk, we shall discuss the critical behavior of the above model. We shall show that the scaled and ordered sizes of connected components converge in distribution to the ordered lengths of excursions above zero of a reflected Brownian motion with a certain drift. The talk is based on a joined work with A. Dembo and S. Vadlamani.

**Kavita Ramanan**: Obliquely reflected diffusions in non-smooth domains

Abstract: Obliquely reflected Brownian motions in smooth domains are classical objects that have been well understood for half a century. Over the last two decades, a theory for obliquely reflected Brownian motions in piecewise smooth domains has been developed, motivated by applications in a variety of fields ranging from mathematical physics to stochastic networks. We provide an overview of this area and then describe recent results on characterization of obliquely reflected Brownian motions in rough planar domains, where it is not a priori clear how to even define the process. We will also discuss some consequences of this characterization.

This talk will include a discussion of various joint works with K. Burdzy, Z.-Q. Chen, W. Kang and D. Marshall.

**Adela Svejda**: Clock processes on infinite graphs and aging in Bouchaud's asymmetric trap model

Abstract: A clock process is the total time that elapses along a given length of the trajectory of a random motion. It is the key object in connection with aging - a phenomenon of random dynamics in random environments whose convergence towards equilibrium becomes slower as time elapses. Based on a method for proving convergence of partial sum processes due to Durrett and Resnick, convergence criteria for clock processes were established for dynamics on finite graphs by Bovier and Gayrard. In this talk, we study dynamics that are defined on infinite graphs and present general convergence criteria for their clock processes. As an application we prove the existence of a normal aging regime in Bouchaud's asymmetric trap model on**Z**^{d}for d ≥ 2.

This talk is based on joint work with V. Gayrard.