Tree limits and limits of random trees

Local limits of random trees and maps

These pages form the lecture notes of a mini-course given at the IXth Young European Probabilists conference that took place in Eindhoven (March 2012). They contain some material extracted from [2, 8, 9]. We survey some aspects of local limits of random trees and maps and deal in particular with the so-called geometric Galton-Watson tree conditioned to survive and the uniform infinite planar qu...

Scaling limits of large random trees

The goal of these lectures is to survey some of the recent progress on the description of largescale structure of random trees. We will use the framework of Markov branching sequences of trees and develop several applications, to combinatorial trees, Galton-Watson trees, some dynamical models of randomly growing trees, cut trees, etc. This is a rough draft – to be completed – all comments are w...

Generalized Covering Groups and Direct Limits

A note on the scaling limits of random Pólya trees

Panagiotou and Stufler (arXiv:1502.07180v2) recently proved one important fact on their way to establish the scaling limits of random Pólya trees: a uniform random Pólya tree of size n consists of a conditioned critical Galton-Watson tree Tn and many small forests, where with probability tending to one as n tends to infinity, each forest Fn(v) is maximally of size |Fn(v)| = O(log n). Their proo...

Limits in modified categories of interest

‎We firstly prove the completeness of the category of crossed modules in a modified category of interest‎. ‎Afterwards‎, ‎we define pullback crossed modules and pullback cat objects that are both obtained by pullback diagrams with extra structures on certain arrows‎. ‎These constructions unify many corresponding results for the cases of groups‎, ‎commutative algebras and can also be adapted to ...