TY - JOUR

T1 - Scaling limits of the Schelling model

AU - Holden, Nina

AU - Sheffield, Scott

N1 - Funding Information:
We thank Omer Tamuz for introducing us to the Schelling model, for numerous helpful discussions, for comments on an earlier draft of this paper, and for allowing us to use his simulations in Fig. . We thank Chris Burdzy for our discussion about his paper [], and we thank Matan Harel for our discussions about Proposition . We thank Nicole Immorlica, Robert Kleinberg, Brendan Lucier, and Rad Niazadeh for discussions about this paper and some of their related work [, ]. The first author was supported by a fellowship from the Norwegian Research Council. The second author was supported by NSF Grants DMS 1209044 and DMS 1712862.
Publisher Copyright:
© 2019, Springer-Verlag GmbH Germany, part of Springer Nature.

PY - 2020/2/1

Y1 - 2020/2/1

N2 - 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- ϕ at zero (taken over all 1-Lipschitz functions 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.

AB - 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- ϕ at zero (taken over all 1-Lipschitz functions 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.

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U2 - 10.1007/s00440-019-00918-0

DO - 10.1007/s00440-019-00918-0

M3 - Article

AN - SCOPUS:85068855184

SN - 0178-8051

VL - 176

SP - 219

EP - 292

JO - Probability Theory and Related Fields

JF - Probability Theory and Related Fields

IS - 1-2

ER -