# Probability and Statistics Seminar

## Upcoming seminars in probability and statistics

**24 October, 2019 **

Details: **Henrik Hult****, **KTH, Ångström 64119, 10:15 - 11:15 am

**Title: Diffusion interpolation in auto encoders**

**Abstract: **Auto encoding models have been extensively studied in computer science. They provide a framework for simulation, as well as for analysing feature learning. Furthermore, they are efficient in performing interpolations between data-points in semantically meaningful ways. We introduce a method for generating stochastic sequences from auto encoders trained on flattened sequences (e.g video sample from auto encoders trained to generate a video frame); as well as a canonical, dimension independent method for generating stochastic interpolations. The methods are based on diffusion bridges and we use the variational auto encoder model for demonstration purposes. This is joint work with Carl Ringqvist, Judith Butepage, and Hedvig Kjellström.

**28 November, 2019 **

Details: **Victor Falgas-Ravry, **Umeå universitet, Ångström 64119, 10:15 - 11:15 am

**Title: On 1-independent random graphs**

**Abstract: **Let *H* be a connected graph. The p-random graph *H _{p}* is obtained by including each edge of

*H*at random with probability

*p*, independently of all the rest. The connectivity properties of

*H*have been widely studied in random graph theory (in particular when

_{p}*H=K*and

_{n}*H*is the Erdős-Rényi random graph model) and in percolation theory (in particular when

_{p}*H*is the square integer lattice ℤ

^{2}).

In this talk, I will be interested in studying the same connectivity properties but in a different class of random graph models, for which there may be some local dependencies between the edges. Formally, a 1-independent model (1-ipm) on *H* is a probability measure *μ* on the subgraphs of *H* such that in a *μ*-random graph *G _{μ}*, events supported on disjoint vertex-sets are independent.

Consider a 1-ipm *G _{μ}* on

*H*in which each edge is present with probability at least p. If

*H*is finite, what can we say about the probability that

*G*is connected? If

_{μ}*H*is infinite, what can we say about the probability that

*G*contains an infinite connected component? I will report some recent (modest) progress on these questions and, time allowing, I will discuss some of the many open problems in the area.

_{μ}Joint work with A. Nicholas Day and Robert Hancock.

**5 December, 2019 **

Details: **Benjamin Thomas Hansen****, **University of Groningen, Ångström 64119, 10:15 - 11:15 am

**Title: Voronoï percolation in the hyperbolic plane**

**Abstract: ** We consider site percolation on Voronoï cells generated by a Poisson point process on the hyperbolic plane H^{2}. Each cell is coloured black independently with probability *p*, otherwise the cell is coloured white. Benjamini and Schramm proved the existence of three phases: For *p* ∈ [0, *p _{c}*] all black clusters are bounded and there is a unique infinite white cluster. For

*p*∈ (

*p*), there are infinitely many unbounded black and white clusters. For

_{c}, p_{u}*p*∈ [

*p*, 1] there is a unique infinite black cluster and all white clusters are bounded. They also showed

_{u}*p*= 1 −

_{u}*p*. The critical values

_{c}*p*and

_{c}*p*depend on the intensity of the Poisson point process. We prove that

_{c}*p*tends to 1/2 as the intensity tends to infinity. This confirms a conjecture of Benjamini and Schramm.

_{c}Joint work with Tobias Müller.