arXiv subjects

Vivian Olsiewski Healey

Publications and source records attributed to Vivian Olsiewski Healey.

7 recordsLinked to original sources

arXiv2026

Large deviations of Dyson Brownian motion on the circle and multiradial SLE(0+)

We show a finite-time large deviation principle (LDP) for "Dyson type" diffusion processes, including Dyson Brownian motion (DBM) on the circle, for a fixed number of particles as the coupling parameter $β=8/κ$ tends to $+\infty$. Zero-energy systems correspond to the Calogero-Moser-Sutherland integrable system. We also characterize the large-time behavior of finite-energy systems: zero-energy systems approach exponentially fast a static equally-spaced configuration, while finite-energy systems may have polynomial convergence rates, and the system may never become static. We use our DBM result to derive a finite-time LDP in the Hausdorff metric for multiradial Schramm-Loewner evolution, SLE$_κ$, as $κ$ tends to $0+$, with good rate function being the multiradial Loewner energy. Using a derivative estimate for the radial Loewner map in terms of the energy of its driving function, we show that finite-energy multiradial Loewner hulls are disjoint unions of simple curves, except at their common endpoint.

math.PR↗

arXiv2026

Multiradial Schramm-Loewner evolution: Infinite-time large deviations and transience

In previous work [AHP24], we proved a finite-time large deviation principle in the Hausdorff metric for multiradial Schramm-Loewner evolution, SLE$(κ)$, as $κ\to 0$, with good rate function being the multiradial Loewner energy. Here, we extend this result to infinite time in the topology of common-capacity-parameterized curves, and streamline the proof. A main step is to derive detailed escape probability estimates for multiradial SLE$(κ)$ curves in the common parameterization, which extend the single-curve estimates achieved in [AP26]. As a by-product, we also get that multiradial SLE$(κ)$ curves, with $κ\leq 8/3$, are transient at their common terminal point, generalizing [FL15, HL21]. As a corollary to the LDP result, we obtain explicit asymptotics of the Brownian loop measure interaction term for finite-energy radial multichords, which is linear in the capacity-time and coincides with a certain choice of a cocycle for the Virasoro algebra.

math.PR↗

arXiv2025

Scaling limits of branching Loewner evolutions and the Dyson superprocess

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.

math.PR↗

arXiv2023

Galois groups and prime divisors in random quadratic sequences

Given a set $S=\{x^2+c_1,\dots,x^2+c_s\}$ defined over a field and an infinite sequence $γ$ of elements of $S$, one can associate an arboreal representation to $γ$, generalizing the case of iterating a single polynomial. We study the probability that a random sequence $γ$ produces a ``large-image'' representation, meaning that infinitely many subquotients in the natural filtration are maximal. We prove that this probability is positive for most sets $S$ defined over $\mathbb{Z}[t]$, and we conjecture a similar positive-probability result for suitable sets over $\mathbb{Q}$. As an application of large-image representations, we prove a density-zero result for the set of prime divisors of some associated quadratic sequences. We also consider the stronger condition of the representation being finite-index, and we classify all $S$ possessing a particular kind of obstruction that generalizes the post-critically finite case in single-polynomial iteration.

math.NT↗

arXiv2022

N-sided Radial Schramm-Loewner Evolution

We use the interpretation of the Schramm-Loewner evolution as a limit of path measures tilted by a loop term in order to motivate the definition of $n$-radial SLE going to a particular point. In order to justify the definition we prove that the measure obtained by an appropriately normalized loop term on $n$-tuples of paths has a limit. The limit measure can be described as $n$ paths moving by the Loewner equation with a driving term of Dyson Brownian motion. While the limit process has been considered before, this paper shows why it naturally arises as a limit of configurational measures obtained from loop measures.

math.PR↗

arXiv2018

Stochastic Canonical Heights

We construct height functions defined stochastically on projective varieties equipped with endomorphisms, and we prove that these functions satisfy analogs of the usual properties of canonical heights. Moreover, we give a dynamical interpretation of the kernel of these stochastic height functions, and in the case of the projective line, we relate the size of this kernel to the Julia sets of the original maps. Finally, as an application, we establish the finiteness of some generalized Zsigmondy sets over global fields.

math.NT↗

arXiv2009

A Family of Recompositions of the Penrose Aperiodic Protoset and Its Dynamic Properties

This paper describes a recomposition of the rhombic Penrose aperiodic protoset due to Robert Ammann. We show that the three prototiles that result from the recomposition form an aperiodic protoset in their own right without adjacency rules. An interation process is defined on the space of Ammann tilings that produces a new Ammann tiling from an existing one, and it is shown that this process runs in parallel to Penrose deflation. Furthermore, by characterizing Ammann tilings based on their corresponding Penrose tilings and the location of the added vertex that defines the recomposition process, we show that this process proceeds to a limit for the local geometry.

math.MG↗