The Schelling model of segregation, introduced by Schelling in 1969 as a model for residential segregation in cities, describes how populations of multiple types self-organize to form homogeneous clusters of one type. In this model, vertices in an N -dimensional lattice are initially assigned types randomly. As time evolves, the type at a vertex v has a tendency to be replaced with the most common type within distance w of v . We present the first mathematical description of the dynamical scaling limit of this model as w tends to infinity and the lattice is correspondingly rescaled. We do this by deriving an integro-differential equation for the limiting Schelling dynamics and proving almost sure existence and uniqueness of the solutions when the initial conditions are described by white noise. The evolving fields are in some sense very “rough” but we are able to make rigorous sense of the evolution. In a key lemma, we show that for certain Gaussian fields h , the supremum of the occupation density of $$h-\phi $$ h - ϕ at zero (taken over all 1-Lipschitz functions $$\phi $$ ϕ ) is almost surely finite, thereby extending a result of Bass and Burdzy. In the one dimensional case, we also describe the scaling limit of the limiting clusters obtained at time infinity, thereby resolving a conjecture of Brandt, Immorlica, Kamath, and Kleinberg.

中文翻译：

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Schelling 模型的缩放限制

Schelling 隔离模型由 Schelling 于 1969 年引入，作为城市住宅隔离的模型，描述了多种类型的人口如何自我组织以形成一种类型的同质集群。在这个模型中，N 维点阵中的顶点最初被随机分配类型。随着时间的推移，顶点 v 处的类型有被 v 距离 w 内最常见的类型替换的趋势。我们提出了这个模型的动态缩放限制的第一个数学描述，因为 w 趋于无穷大，并且相应地重新缩放晶格。我们通过推导限制 Schelling 动力学的积分微分方程并证明当初始条件由白噪声描述时几乎肯定的存在性和唯一性来做到这一点。不断发展的领域在某种意义上非常“粗糙”，但我们能够对演变做出严格的理解。在一个关键引理中，我们证明对于某些高斯场 h ，$$h-\phi $$ h - ϕ 的占据密度在零时的上限值（采用所有 1-Lipschitz 函数 $$\phi $$ ϕ ）几乎肯定是有限的，从而扩展了 Bass 和 Burdzy 的结果。在一维情况下，我们还描述了在无限时间获得的极限簇的标度极限，从而解决了 Brandt、Immorlica、Kamath 和 Kleinberg 的猜想。