Brandeis

I will discuss the percolation model on planar triangulations, and present a bijection that is key to relating this model to some fundamental probabilistic objects. I will attempt to achieve several goals:
1. Present the site-percolation model on random planar triangulations.
2. Provide an informal introduction to several probabilistic objects: the Gaussian free field, Schramm-Loewner evolutions, and the Brownian map.
3. Present a bijective encoding of percolated triangulations by certain lattice paths, and explain its role in establishing exact relations between the above-mentioned objects.
This is joint work with Nina Holden, and Xin Sun.

Morgan and Culler proved in the 1980’s that a minimal action of a free group on a tree is

completely determined by its length function. This theorem has been of fundamental importance in the

study of automorphisms of free groups. In particular, it gives rise to a compactification of Culler-Vogtmann's

Outer Space. We prove a 2-dimensional analogue of this theorem for right-angled Artin groups acting on

CAT(0) rectangle complexes. (Joint work with M. Margolis)