Tree Embedding via the Loewner Equation and the Dyson
Superprocess
In its most well-known form, the Loewner equation gives a
correspondence between curves in the upper half-plane and continuous
real functions (called the "driving function" for the equation). We
consider the generalized Loewner equation, where the driving function
has been replaced by a time-dependent real measure. In the first part
of the talk, we investigate the delicate relationship between the
driving measure and the generated hull. We show that certain discrete
driving measures (closely related to branching Dyson Brownian motion)
generate tree embeddings. In the second part of the talk, we describe
the superprocess that is the scaling limit of branching Dyson Brownian
motion when the underlying (critical, binary) Galton-Watson trees are
conditioned to converge to the continuum random tree.