Scaling limits of branching Loewner evolutions and the Dyson superprocess

TL;DR

This paper develops a novel conformal process combining branching processes with Loewner evolution, analyzing its scaling limits and embedding properties, and introduces the Dyson superprocess as a free probability analogue of classical superprocesses.

Contribution

It introduces a new construction of conformal processes driven by Dyson Brownian motion and establishes their scaling limits as superprocesses, including the Dyson superprocess.

Findings

The hull of the Loewner evolution embeds the genealogical tree with prescribed angles.

Scaling limits of the driving measure are characterized as superprocesses.

The Dyson superprocess is introduced as a free probability analogue of classical superprocesses.

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.