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| Natura: | Preprint |
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2023
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| Accesso online: | https://arxiv.org/abs/2310.02858 |
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| _version_ | 1866908485716803584 |
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| author | Healey, Vivian Olsiewski Menon, Govind |
| author_facet | Healey, Vivian Olsiewski Menon, Govind |
| contents | 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 $β\in (0,\infty]$ between birth and death events.
When $β=\infty$, 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, $\{θ_v \in (0,π/2)\}_{v \in \mathcal{T}}$ the hull of the Loewner evolution is an embedding of $\mathcal{T}$ into the upper half-plane with trivalent edges that meet at angles $(2θ_v,2π-4θ_v,2θ_v)$ at the image of each edge $v$.
We also study the scaling limit when $β\in (0,\infty]$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_02858 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Scaling limits of branching Loewner evolutions and the Dyson superprocess Healey, Vivian Olsiewski Menon, Govind Probability 60J67, 60B20, 60J68 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 $β\in (0,\infty]$ between birth and death events. When $β=\infty$, 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, $\{θ_v \in (0,π/2)\}_{v \in \mathcal{T}}$ the hull of the Loewner evolution is an embedding of $\mathcal{T}$ into the upper half-plane with trivalent edges that meet at angles $(2θ_v,2π-4θ_v,2θ_v)$ at the image of each edge $v$. We also study the scaling limit when $β\in (0,\infty]$ 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. |
| title | Scaling limits of branching Loewner evolutions and the Dyson superprocess |
| topic | Probability 60J67, 60B20, 60J68 |
| url | https://arxiv.org/abs/2310.02858 |