Scaling limits of branching Loewner evolutions and the Dyson superprocess

Abstract

This work introduces a construction of conformal processes that combines the theory of branching processes with chordal Loewner evolution. The main novelty lies in the choice of driving measure for the Loewner evolution: given a finite genealogical tree $\mathcal{T}$, we choose a driving measure for the Loewner evolution that is supported on a system of particles that evolves by Dyson Brownian motion at inverse temperature $\beta \in (0,\infty ]$ between birth and death events. When $\beta =\infty$, the driving measure degenerates to a system of particles that evolves through Coulombic repulsion between branching events. In this limit, the following graph embedding theorem is established: When $\mathcal{T}$ is equipped with a prescribed set of angles, $\{{\theta }_v\in (0,\pi /2){\}}_{v\in \mathcal{T}}$ the hull of the Loewner evolution is an embedding of $\mathcal{T}$ into the upper half-plane with trivalent edges that meet at angles $(2{\theta }_v,2\pi -4{\theta }_v,2{\theta }_v)$ at the image of each edge $v$. We also study the scaling limit when $\beta \in (0,\infty ]$ is fixed and $\mathcal{T}$ is a binary Galton-Watson process that converges to a continuous state branching process. We treat both the unconditioned case (when the Galton-Watson process converges to the Feller diffusion) and the conditioned case (when the Galton-Watson tree converges to the continuum random tree). In each case, we characterize the scaling limit of the driving measure as a superprocess. In the unconditioned case, the scaling limit is the free probability analogue of the Dawson-Watanabe superprocess that we term the Dyson superprocess.

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