{"title":"Stability of Regularized Hastings–Levitov Aggregation in the Subcritical Regime","authors":"James Norris, Vittoria Silvestri, Amanda Turner","doi":"10.1007/s00220-024-04960-5","DOIUrl":null,"url":null,"abstract":"<p>We prove bulk scaling limits and fluctuation scaling limits for a two-parameter class ALE<span>\\(({\\alpha },\\eta )\\)</span> of continuum planar aggregation models. The class includes regularized versions of the Hastings–Levitov family HL<span>\\(({\\alpha })\\)</span> and continuum versions of the family of dielectric-breakdown models, where the local attachment intensity for new particles is specified as a negative power <span>\\(-\\eta \\)</span> of the density of arc length with respect to harmonic measure. The limit dynamics follow solutions of a certain Loewner–Kufarev equation, where the driving measure is made to depend on the solution and on the parameter <span>\\({\\zeta }={\\alpha }+\\eta \\)</span>. Our results are subject to a subcriticality condition <span>\\({\\zeta }\\leqslant 1\\)</span>: this includes HL<span>\\(({\\alpha })\\)</span> for <span>\\({\\alpha }\\leqslant 1\\)</span> and also the case <span>\\({\\alpha }=2,\\eta =-1\\)</span> corresponding to a continuum Eden model. Hastings and Levitov predicted a change in behaviour for HL<span>\\(({\\alpha })\\)</span> at <span>\\({\\alpha }=1\\)</span>, consistent with our results. In the regularized regime considered, the fluctuations around the scaling limit are shown to be Gaussian, with independent Ornstein–Uhlenbeck processes driving each Fourier mode, which are seen to be stable if and only if <span>\\({\\zeta }\\leqslant 1\\)</span>.</p>","PeriodicalId":522,"journal":{"name":"Communications in Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":2.2000,"publicationDate":"2024-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Communications in Mathematical Physics","FirstCategoryId":"101","ListUrlMain":"https://doi.org/10.1007/s00220-024-04960-5","RegionNum":1,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"PHYSICS, MATHEMATICAL","Score":null,"Total":0}
引用次数: 0
Abstract
We prove bulk scaling limits and fluctuation scaling limits for a two-parameter class ALE\(({\alpha },\eta )\) of continuum planar aggregation models. The class includes regularized versions of the Hastings–Levitov family HL\(({\alpha })\) and continuum versions of the family of dielectric-breakdown models, where the local attachment intensity for new particles is specified as a negative power \(-\eta \) of the density of arc length with respect to harmonic measure. The limit dynamics follow solutions of a certain Loewner–Kufarev equation, where the driving measure is made to depend on the solution and on the parameter \({\zeta }={\alpha }+\eta \). Our results are subject to a subcriticality condition \({\zeta }\leqslant 1\): this includes HL\(({\alpha })\) for \({\alpha }\leqslant 1\) and also the case \({\alpha }=2,\eta =-1\) corresponding to a continuum Eden model. Hastings and Levitov predicted a change in behaviour for HL\(({\alpha })\) at \({\alpha }=1\), consistent with our results. In the regularized regime considered, the fluctuations around the scaling limit are shown to be Gaussian, with independent Ornstein–Uhlenbeck processes driving each Fourier mode, which are seen to be stable if and only if \({\zeta }\leqslant 1\).
我们证明了连续平面聚集模型的双参数类 ALE\(({\alpha },\eta )\) 的体量缩放极限和波动缩放极限。该类包括黑斯廷斯-列维托夫模型族 HL\(({\alpha })\)的正则化版本和介电分解模型族的连续化版本,其中新粒子的局部附着强度被指定为弧长密度相对于谐波度量的负幂次\(-\ea \)。极限动力学遵循某个卢瓦纳-库法列夫方程的解,其中驱动度量取决于解和参数\({\zeta }={\alpha }+\eta \)。我们的结果受制于一个亚临界条件(\({\zeta }\leqslant 1\): 这包括HL(({\alpha })\) for \({\alpha }\leqslant 1\) and also the case \({\alpha }=2,\eta =-1\) corresponding to a continuum Eden model.黑斯廷斯和列维托夫预测了在\({\alpha }=1\) 时 HL\(({\alpha })\) 的行为变化,这与我们的结果一致。在所考虑的正则化机制中,围绕缩放极限的波动被证明是高斯的,每个傅里叶模式都有独立的奥恩斯坦-乌伦贝克过程驱动,只有当\({\zeta }\leqslant 1\) 时,这些过程才是稳定的。
期刊介绍:
The mission of Communications in Mathematical Physics is to offer a high forum for works which are motivated by the vision and the challenges of modern physics and which at the same time meet the highest mathematical standards.