Math Colloquium

### Speaker: Jakob Björnberg, Chalmers Tekniska Högskola, Göteborg

Title: Percolation in random quadrangulations of the half-plane

Location: V-157, VRII

Time: Monday, November 17 at 15:00-16:00.

### Abstract:

The topic of random planar quadrangulations (and, more generally, random planar maps) is of interest to probabilists, combinatorialists and physicists alike. Recent years have seen considerable progress on understanding large random planar maps themselves, and the next big challenge is to understand maps “with matter”; that is, to study models from statistical physics on large random planar maps. In this talk we consider such a model, specifically site percolation on uniform quadrangulations of the half-plane. The talk is based on ongoing work together with Sigurdur Stefansson (Reykjavik) where we use a “spatial Markovian property” of the quadrangulations (first described by O. Angel) to study the percolation phase transition. We will describe this Markovian property, see how Angel (et al) used it in the case of triangulations, and discuss results and challenges for the case of quadrangulations.

Math Colloquium

### Speaker: Sigurður Örn Stefánsson, University of Iceland

Title: Convergence of random planar maps to the Brownian tree

Location: V-157, VRII

Time: Monday, October 27 at 15:15-16:15.

### Abstract:

Random planar maps are defined by assigning non-negative weights to each face of a planar map and the weight of a face depends only on its degree. I will explain the Bouttier-Di Francesco-Guitter bijection between the planar maps and a class of labelled trees called mobiles. By throwing away labels one can, via another bijection, relate the mobiles to the model of so-called simply generated trees which are understood in detail. For certain choices of weights a unique large face, having degree proportional to the total number of edges in the maps, appears with high probability when the maps are large. This corresponds to a recently studied phenomenon of condensation in simply generated trees where a vertex having degree proportional to the size of the trees appears. In this case the planar maps, with a properly rescaled graph metric, are shown to converge in distribution towards Aldous’ Brownian tree in the Gromov-Hausdorff topology.

Math Colloquium

### Speaker: Thomas Vallier, University of Iceland

Title: Bootstrap percolation on the random graph \(G_{n,p}\)

Location: V-157, VRII

Time: Monday September 15, at 15:00-16:00.

### Abstract:

Bootstrap percolation on the random graph \(G_{n,p}\) is a process of spread of “activation” on a given realization of the graph with a given number of initially active nodes. At each step those vertices which have not been active but have at least \(r ≥ 2\) active neighbours become active as well.

We consider the n vertices with global connections inherited from the structure of the graph \(G_{n,p}\), meaning that any two vertices share an edge with probability \(p\) independently of the others.

The presentation is based on the article ”Bootstrap percolation on the random graph \(G_{n,p}\)” by Janson, Luczak, Turova and Vallier.

Among other results, they study the size \(A^*\) of the final active set depending on the number of vertices active at the origin as a function of n (the number of vertices) and \(p = p(n)\) (the probability of connections) which is written \(a_0(n, p) = a_0\). The model exhibits a sharp phase transition: depending on the parameters of the model the final size of activation with a high probability is either \(n − o(n)\) or it is \(o(n)\).

I will give a pictorial introduction to the model and explain briefly the approach of the authors to derive the threshold for bootstrap percolation on \(G_{n,p}\).