Some properties of discrete stable maps
Vortrag von Dr. Loïc Richier
Sprecher eingeladen von: Prof. Dr. Jean Bertoin
Datum: 21.03.18 Zeit: 17.15 - 19.00 Raum: Y27H12
The purpose of this talk is to discuss some properties of random planar maps such that the degree of a typical face falls within the domain of attraction of a stable distribution with parameter α∈(1,2). These maps, that we call discrete stable, have attracted a lot of attention. In 2011, Le Gall and Miermont proved that discrete stable maps admit subsequential scaling limits, suggesting the existence of a stable counterpart to the Brownian map. At the same time, Borot, Bouttier and Guitter established a connection between discrete stable maps and the O(n) loop models on planar maps.
In the first part of the talk, we will investigate the scaling limits of large faces of discrete stable maps (or, equivalently, large loops in the O(n) model). The motivation comes from a conjecture stating that these large faces are self-intersecting in the so-called dense regime α∈(1,3/2), and self-avoiding in the so-called dilute regime α∈(3/2,2). In the second part, we will deal with the bond percolation model on discrete stable maps in the dilute regime. We will discuss a duality property showing that at criticality, the open percolation cluster of the origin is itself a discrete stable map in the dense regime, with explicit parameter. This result is inspired by recent work of Miller, Sheffield and Werner, who established such a duality property in Conformal Loop Ensembles. To conclude, we will mention other results concerning the bond percolation model, such as the scaling limits of percolation clusters and the sharpness of the phase transition. Partially joint work with Nicolas Curien and Igor Kortchemski.