Abstract:

Consider a binary tree with n labeled leaves. Randomly select
a leaf for removal and then reinsert it back on an edge selected at
random from the remaining structure. This produces a Markov chain on the
space of n-leaved binary trees whose invariant distribution is the
uniform distribution. David Aldous, who introduced and analyzed this
Markov chain, conjectured the existence of a continuum limit of this
process if we remove labels from leaves, scale edge-length and time
appropriately with n, and let n go to infinity. The conjectured
diffusion will have an invariant distribution given by the so-called
Brownian Continuum Random Tree. In a series of papers, co-authored with
N. Forman, D. Rizzolo, and M. Winkel, we construct this continuum limit.
This talk will be an overview of our construction and describe the path
behavior of this limiting object.