Málstofa í stærðfræði
Fyrirlesari: Sigurður Örn Stefánsson, Háskóla Íslands
Titill: Random maps with large faces
Staðsetning: Tg-227 í Tæknigarði.
Tímasetning: Fimmtudaginn 31. mars 2022, kl. 10:30.
There has been an immense progress in the understanding of random planar maps in the last two decades. An important breakthrough was the independent proofs of Le Gall and Miermont that certain classes of these maps (uniform triangulations and uniform 2p-angulations) converge towards the so called Brownian map. Subsequently there have been many extensions showing that the Brownian map arises as a universal limit of a large family of discrete models. Another important family of random maps are the so called stable maps which arise as limits of random planar maps which are defined in such a way that large faces form in the maps. The study of stable maps is motivated by the conjecture (and in some cases proven fact) that they appear as natural objects when the Brownian map is decorated with statistical mechanical models. To date much less is known about the stable maps than the Brownian map, although there are some exciting results on the horizon.
The focus of the current talk is a model of causal planar maps which was introduced in its original form by Ambjorn and Loll. The limit of the causal maps in the uniform case (which is analogous to the Brownian map case above) turns out to be trivial. However when the measure is tweaked so that large faces are forced to appear, we show that there arises an interesting scaling limit which we call the stable shredded sphere. I will define the stable shredded sphere, describe some of its properties and explain briefly the key ingredients in the proof of the limit result.
This is joint work with Jakob Björnberg and Nicolas Curien. See https://arxiv.org/abs/1912.01378 for details.