平面内扩展源DLA的收敛速率研究

IF 1 3区数学 Q1 MATHEMATICS

Potential Analysis Pub Date : 2023-10-16 DOI:10.1007/s11118-023-10102-8

David Darrow

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引用次数: 2

摘要

内部DLA (IDLA)是一种内部聚集模型，其中粒子依次从原点随机行走，并在到达一个未被占用的位置时停止。Levine和Peres表明，当粒子从固定的多点分布开始时，改进的IDLA过程具有与特定障碍问题相关的确定性缩放限制。在本文中，我们研究了这种“扩展源”IDLA在平面上的收敛速度到它的缩放极限。我们证明，当$$\delta $$ δ为晶格尺寸时，IDLA占位集的波动距离其标度极限最多为$$\delta ^{3/5}$$ δ 3 / 5阶，且概率至少为$$1-e^{-1/\delta ^{2/5}}$$ 1 - e - 1 / δ 2 / 5。

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A Convergence Rate for Extended-Source Internal DLA in the Plane

Abstract Internal DLA (IDLA) is an internal aggregation model in which particles perform random walks from the origin, in turn, and stop upon reaching an unoccupied site. Levine and Peres showed that, when particles start instead from fixed multiple-point distributions, the modified IDLA processes have deterministic scaling limits related to a certain obstacle problem. In this paper, we investigate the convergence rate of this “extended source” IDLA in the plane to its scaling limit. We show that, if $$\delta $$ δ is the lattice size, fluctuations of the IDLA occupied set are at most of order $$\delta ^{3/5}$$ δ 3 / 5 from its scaling limit, with probability at least $$1-e^{-1/\delta ^{2/5}}$$ 1 - e - 1 / δ 2 / 5 .

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来源期刊

Potential Analysis 数学-数学

CiteScore

2.20

自引率

9.10%

发文量

审稿时长

>12 weeks

期刊介绍： The journal publishes original papers dealing with potential theory and its applications, probability theory, geometry and functional analysis and in particular estimations of the solutions of elliptic and parabolic equations; analysis of semi-groups, resolvent kernels, harmonic spaces and Dirichlet forms; Markov processes, Markov kernels, stochastic differential equations, diffusion processes and Levy processes; analysis of diffusions, heat kernels and resolvent kernels on fractals; infinite dimensional analysis, Gaussian analysis, analysis of infinite particle systems, of interacting particle systems, of Gibbs measures, of path and loop spaces; connections with global geometry, linear and non-linear analysis on Riemannian manifolds, Lie groups, graphs, and other geometric structures; non-linear or semilinear generalizations of elliptic or parabolic equations and operators; harmonic analysis, ergodic theory, dynamical systems; boundary value problems, Martin boundaries, Poisson boundaries, etc.