TY - BOOK

T1 - Degenerate regimes for random growth models in the complex plane

AU - Higgs, Frankie

PY - 2022

Y1 - 2022

N2 - Diffusion-limited aggregation (DLA) is among the most studied models in mathematical physics, and obtaining results on the limiting geometry has proved one of the more difficult problems of the past few decades.At the same time, techniques from complex analysis and conformal mapping theory have become popular following the definition of the Schramm-Loewner evolution by Schramm in 2000 and its subsequent use to solve scaling limit and other problems for planar models in statistical physics.This thesis analyses the aggregate Loewner evolution (ALE) model, introduced by Sola, Turner and Viklund in 2018 to generalise versions of DLA in the complex plane. The ALE is a model of growth where a particle is added at a location on the existing cluster at a point chosen by a regularised version of harmonic measure, transformed by a parameter η. The three main chapters of this thesis examine ALE for extreme values of η, where the behaviour becomes degenerate in some sense. In Chapter 2, we demonstrate that for large negative values, η < −2, which correspond to attachment in areas of low harmonic measure, each particle is attached near the base of the previous particle. A consequence of this is the convergence of the ALE cluster to a Schramm-Loewner evolution  curve. This contributes one of the first scaling limit results with a non-deterministic limit for an aggregation model in the complex plane.In Chapter 3, we extend the results of Sola, Turner and Viklund for η > 1, demonstrating that when started from a non-trivial initial configuration, the scaling limit is the geodesic Laplacian path model, a model of needle growth generalising several physical models.In Chapter 4 we examine stability of the ALE for η > 1. We find a phase transition, with increasing stability such that an additional small  perturbation survives if and only if 1 < η < 2.

AB - Diffusion-limited aggregation (DLA) is among the most studied models in mathematical physics, and obtaining results on the limiting geometry has proved one of the more difficult problems of the past few decades.At the same time, techniques from complex analysis and conformal mapping theory have become popular following the definition of the Schramm-Loewner evolution by Schramm in 2000 and its subsequent use to solve scaling limit and other problems for planar models in statistical physics.This thesis analyses the aggregate Loewner evolution (ALE) model, introduced by Sola, Turner and Viklund in 2018 to generalise versions of DLA in the complex plane. The ALE is a model of growth where a particle is added at a location on the existing cluster at a point chosen by a regularised version of harmonic measure, transformed by a parameter η. The three main chapters of this thesis examine ALE for extreme values of η, where the behaviour becomes degenerate in some sense. In Chapter 2, we demonstrate that for large negative values, η < −2, which correspond to attachment in areas of low harmonic measure, each particle is attached near the base of the previous particle. A consequence of this is the convergence of the ALE cluster to a Schramm-Loewner evolution  curve. This contributes one of the first scaling limit results with a non-deterministic limit for an aggregation model in the complex plane.In Chapter 3, we extend the results of Sola, Turner and Viklund for η > 1, demonstrating that when started from a non-trivial initial configuration, the scaling limit is the geodesic Laplacian path model, a model of needle growth generalising several physical models.In Chapter 4 we examine stability of the ALE for η > 1. We find a phase transition, with increasing stability such that an additional small  perturbation survives if and only if 1 < η < 2.

U2 - 10.17635/lancaster/thesis/1846

DO - 10.17635/lancaster/thesis/1846

M3 - Doctoral Thesis

PB - Lancaster University

ER -