# Graphs, Geometry and Probability Workshop

### Thursday 28 November 2013

All talks will be held in room MS.04.

Organiser: Agelos Georgakopoulos

### Programme

2.05 - 2.55 Nathanael Berestycki Condensation of random walks and Wulff crystal

Abstract: We introduce a Gibbs measure on nearest-neighbour paths of length *t* in the Euclidean *d*-dimensional lattice, where each path is penalized by a factor proportional to the size of its boundary and an inverse temperature β. The resulting random shape can be thought of as a random walk construction of the Wulff crystal, representing the distribution of a diluted polymer in a poor solvent. We prove that, for all β > 0, the random walk condensates to a set of diameter *t*^{1/(d+1)} with high probability. We further speculate that the limiting shape shares the same exponents as the KPZ universality class. A similar result holds for a random walk conditioned to have local time greater than β everywhere in its range, provided that β is greater than some explicit constant (the log of the connective constant in dimension 2). This answers a question of Itai Benjamini.Joint work with Ariel Yadin.

2.55 - 3.15 Coffee in the Mathematics Institute Common Room

3.15 - 4.05 Imre Leader Euclidean Ramsey Theory

Abstract: A finite subset of Euclidean space* R ^{d} *is called Ramsey if, for any

*k*, there exists an

*n*such that whenever

*R*is partitioned into

^{n}*k*classes (or 'is

*k*-coloured') there is a copy of our set in one of the classes. Which sets are Ramsey?The talk will start with background, and will proceed on to some more recent developments. No knowledge of Ramsey theory will be assumed.

4.15 - 5.00 Agelos Georgakopoulos Discrete Riemann mapping and the Poisson boundary

Abstract: Answering a question of Benjamini & Schramm, we show that the Poisson boundary of any planar, uniquely absorbing (e.g. one-ended and transient) graph with bounded degrees can be realised geometrically as a circle, which circle arises from a discrete version of Riemann's mapping theorem. I will start by explaining this discrete Riemann map, drawing various analogies between the discrete and continuous case.The talk will be self-contained, assuming no prior knowledge of the mentioned terminology, and will contain many pictures.

5:30 Dinner in the Mathematics Institute Common Room

If you are interested to participate in the dinner please contact Agelos.

The meeting is supported by an LMS Conference grant to celebrate new appointments. The appointment being 'celebrated' is that of Agelos Georgakopoulos.