Jomy Alappattu
UC Berkeley
Fragmentation and coalescence of conditioned Galton-Watson forests
Abstract: Given a Galton-Watson forest of k trees with some offspring distribution conditioned to have n total vertices, suppose that an edge is deleted uniformly at random from this forest. For what offspring distributions is it true that the resulting forest is distributed like a Galton-Watson forest of k+1 trees with the same offspring distribution conditioned to have n vertices?

This question is related to recent work on fragmentation of partitions of the set [n] = {1,2,...,n}. After reviewing the known results about partitions, we show how to use generating functions to pass from partitions to forests and determine the offspring distributions for which this fragmentation model works. We will also discuss certain interesting combinatorial interpretations of some of the offspring distributions. Finally, we describe how to go in the reverse direction, and determine how trees in a conditioned Galton-Watson forest should coalesce in order to preserve the conditioned structure.

This is joint work with N. Berestycki and J. Pitman.