Math Colloquium

### Speaker: Adam Timar, Renyi Institute, Budapest

Title: Allocation rules for the Poisson point process

Location: Árnagarður 101.

Time: Friday, June 3, at 13:20-14:20.

### Abstract:

Consider the Poisson point process in Euclidean space. We are interested in functions on this random point set whose value in each configuration point is given by some “local” rule (no “central planning”). One example is the so-called allocation problem, where we want to partition R^d to sets of measure 1 and match them with the point process, in a translation equivariant way. We want to make the allocated set optimal in some sense (e.g., the distribution of the diameter shows fast decay). We will present some allocation schemes, among them one with an optimal tail, which is joint work with R. Marko.

Math Colloquium

### Speaker: Jakob Björnberg

Title: Random permutations and quantum Heisenberg models

Location: V-157, VRII.

Time: Friday, April 1 at 13:20.

### Abstract:

The interchange process (or random-transposition random walk) is a model for random permutations which is closely related to a model from quantum statistical physics (the ferromagnetic Heisenberg model). In fact, certain ‘cycle-weighted’ interchange processes are equivalent to the latter, and in this talk we present results on such processes. Magnetic ordering in the physical model translates to the occurrence of large cycles in the random permutation.

We focus on the case when the underlying graph is the complete graph (i.e. the ‘mean-field’ case in physical jargon). By a combination of probabilistic techniques and some group character theory we can obtain nice formulas for expectation values in the model, and then use these to identify the critical point.

Math Colloquium

### Speaker: François David

Title: Planar maps, circle patterns and 2D gravity

Location: V-157, VRII.

Time: Friday, March 11 at 13:20.

### Abstract:

I present a model of random planar triangulations (planar maps) based on circle patterns and discuss its properties. It exemplifies the relations between discrete random geometries in the plane, conformally invariant point processes and two dimensional quantum gravity (Liouville theory and topological gravity).

Math Colloquium

### Speaker: Guenter Last

Title: Second order properties of the Boolean model and the Gilbert graph

Location: V-157, VRII.

Time: Friday, October 30, at 15:00-16:00.

### Abstract:

The Boolean model is a fundamental model of stochastic geometry and continuum percolation. It is a random subset of Euclidean space that arises as the union of random convex grains, independently centered around the points of a stationary Poisson process. The restriction of the Boolean model to a convex and compact observation is a finite union of convex sets. Therefore it makes sense to talk about its intrinsic volumes as volume, surface content, and Euler characteristic.

In this lecture we shall first discuss classical formulae for the densities (normalized expectations) of these intrinsic volumes. Then we proceed with studying asymptotic covariances for growing observation window. These covariances can be expressed in terms of curvature measures associated with a typical grain. In the two-dimensional isotropic case the formulae become surprisingly explicit. We also present a multivariate central limit theorem including Berry-Esseen bounds, derived with the so-called Stein-Malliavin method. If time permits we will also discuss some cluster properties of the Gilbert graph which is a close relative of the Boolean model.

Large parts of the talk are based on joint work with Daniel Hug and

Matthias Schulte.

Math Colloquium

### Speaker: Páll Melsted

Title: Space Utilization of Cuckoo Hashtables

Location: V-157, VRII.

Time: Friday, October 23, at 15:00-16:00.

### Abstract:

We study the space requirements for Cuckoo Hashing. This can be reduced to

the following question in Random Graphs.

We are given two disjoint sets L,R with |L|=n=alpha*m and |R|=m. We construct a random graph G by allowing each x in L to choose d random neighbours in R. The problem is to find m(G), the size of the largest matching in G.

From the point of view of Cuckoo Hashing, a key question is to locate the threshold for when m(G)=n with high probability, since if m(G) < n it is impossible to store all items. We answer this question exactly for all values of d.

Math Colloquium

### Speaker: Hermann Thorisson, University of Iceland

Title: Mass-Stationarity, Shift-Coupling, and Brownian Motion

Location: Naustið, Endurmenntun (here)

Time: Thursday, May 28, at 15:00-16:00.

### Abstract:

After considering mass-stationarity and shift-coupling briefly in an abstract setting, we focus on the special case of stochastic processes on the line associated with diffuse random measures. The main examples are Brownian motion and the Brownian bridge.

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 \(G_{n,p}\).