MARK KAC SEMINAR

Random graphs on surfaces (maps) are natural discrete models of random planar geometries. It was recently shown in a joint work with Bettinelli that a properly rescaled random quadrangulation with a boundary (that is a dissection of a disk into a family of quadrangles) converges to a limiting random metric space, the Brownian disk. The scaling regime in this case is very specific: namely, the perimeter should scale as n^1/2, where n is the the number of quadrangles, and the distances should be rescaled by n^1/4. Other limiting results were obtained in the case of small or large perimeters.

In this work, we investigate all the other possible regimes, in particular the cases where we rescale by a quantity which is much smaller than the typical distances, so that the limiting spaces are non-compact. This is reminiscent of the introduction of the so-called Brownian plane by Curien and Le Gall, which can be seen as the tangent space of the Brownian map near a typical point. By rescaling near a boundary point, we construct in particular a half-plane analog of the Brownian plane. A bit surprisingly though, some other exotic regimes turn out to emerge, including a one-parameter family of spaces that interpolate between the Brownian half-plane and the non-compact version of the Brownian continuum random tree.

This is based on work in collaboration with Erich Baur (ENS de Lyon), Jérémie Bettinelli (Ecole Polytechnique), Emmanuel Jacob (ENS de Lyon) and Gourab Ray (Cambridge).

In this talk I will give an overview about some old and new results on general (Crump-Mode-Jagers type) branching processes (BP) with infinite mean offspring distribution. In particular, I will investigate how to tell whether a BP produces infinitely many offspring in finite time (these are called explosive BPs), especially when there are dependencies between the birth-time of the children of an individual. Natural examples of these arise from epidemic models where individuals are only contagious in a possibly random interval after being infected.

I will spend the other half of the talk on studying information diffusion and weighted distances in the configuration model, in the regime where the degree distribution is a power-law with exponent between (2,3). Here, the local neighborhood of a vertex and the initial stages of the spreading can be approximated by an infinite mean offspring BP, thus providing the motivation of and connection to the first part of the talk. The second part is joint work with Enrico Baroni and Remco van der Hofstad.