On random trees and forests

¹ Laboratoire de Mathématiques d’Orsay, 91405 Orsay, France
² CERMICS, Ecole des Ponts, France
³ LmB, UMR 6623, Université Bourgogne Franche-Comté, CNRS, F-25000 Besançon, France
⁴ CMAP, École Polytechnique, 91128 Palaiseau CEDEX, France
⁵ CERMICS, Ecole des Ponts, France

Abstract

The first talk at the session Random trees and random forests “Journée MAS” (27/08/2021) was presented by I. Kortchemski. After a general up-to-date introduction to local and scaling limits of Bienaymé trees (which are discrete branching trees), he presented new results on precise behavior of the largest out-degree of large branching trees when the offspring distribution μ is subcritical with μ(n) of order n^−β for large n and β > 2 or critical with μ(n) of order n⁻². In the next talk, M. Nassif gave asymptotics of additive functionals of large Bienayme trees in the global regime, which can be understood using scaling limits. Looking at Cayley trees with fixed size, A. Contat established a surprising identity for randomly built independent sets. Eventually J.-J. Duchamps presented some results on the distribution of the discrete Moran forest, a random graph arising in a classical population model at equilibrium.