Pattern Occurrence Counts in Random Planar Maps
Random planar maps have been studied from various aspects during the last 15 or 20 years, including various limiting distributions for several parameters of interest (such as the largest 2-connected component) and local Benjamini-Schramm limits as well as scaling limits. A pattern is a given planar map and we say that it appears in another map if it could be “cut out” just leaving a face. The simplest pattern is just a k-gon. It directly follows from the Benjamini-Schramm limit that the expected number of occurrences of a given pattern is asymptotically linear in the number of edges of the random map. However, it is a challenging problem to provide a more precise limit law. The purpose of this talk is to give a survey on the results and methods that have used so far in order to settle this question. It is conjectured that there is always a central limit theorem – and all results so far support this conjecture.
Date: 23 January 2023, 14:00 (Monday, 2nd week, Hilary 2023)
Venue: Mathematical Institute, Woodstock Road OX2 6GG
Venue Details: L3
Speaker: Michael Drmota (TU Wien)
Organising department: Department of Statistics
Organisers: Matthias Winkel (Department of Statistics, University of Oxford), Julien Berestycki (University of Oxford), Christina Goldschmidt (Department of Statistics, University of Oxford), James Martin (Department of Statistics, University of Oxford)
Part of: Probability seminar
Booking required?: Not required
Audience: Public
Editor: Christina Goldschmidt