Tipping Points in 1-Dimensional Schelling Models with Switching Agents

Barmpalias, George; Elwes, Richard; Lewis-Pye, Andrew

doi:10.1007/s10955-014-1141-5

Cited by 15 publications

(61 citation statements)

References 24 publications

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“…In [3] the authors gave a more general analysis of the same model for τ ∈ [0, 1]. In [4], the authors then analysed variants of the model under Glauber dynamics (as well as variations thereupon), with an additional innovation: the two types of agent may have unequal intolerances τ α , τ β ∈ [0, 1]. We extend the analysis of this phenomenon in the current paper.…”

Section: Introductionmentioning

confidence: 86%

“…The lack of vacancies is arguably a closer approximation to life in modern cities without large numbers of uninhabited properties, and has proved itself more amenable to mathetatical analysis. Numerous variants of Young's model have subsequently been investigated, notably in the work of Zhang [26,27,28], as well as the series of papers of which the current work is a part [6,3,4].…”

Section: Introductionmentioning

confidence: 99%

“…Young's pairwise-swapping dynamic is known to statistical physicists as the Kawasaki dynamic. An even simpler choice, adopted in [4] and the current paper, is the Glauber dynamic, whereby at each time-step a single randomly selected unhappy agent switches type. Although simple, this is by no means unintuitive, the idea is that an agent who becomes unhappy moves out of the city, and is replaced by an agent of the opposite type.…”

Section: Introductionmentioning

confidence: 99%

“…Despite extensive study, unperturbed Schelling models have largely resisted rigorous analysis, prior results generally focusing on variants in which noise is introduced into the dynamics, the resulting system being amenable to standard techniques from statistical mechanics or stochastic evolutionary game theory [25]. A series of recent papers [6,3,4], has seen the first rigorous analyses of 1-dimensional unperturbed Schelling models, in an asymptotic framework largely unknown in statistical mechanics. Here we provide the first such analysis of 2-and 3-dimensional unperturbed models, establishing most of the phase diagram, and answering a challenge from [6].…”

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confidence: 99%

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Unperturbed Schelling Segregation in Two or Three Dimensions

2016

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Schelling's models of segregation, first described in 1969 [18] are among the best known models of self-organising behaviour. Their original purpose was to identify mechanisms of urban racial segregation. But his models form part of a family which arises in statistical mechanics, neural networks, social science, and beyond, where populations of agents interact on networks. Despite extensive study, unperturbed Schelling models have largely resisted rigorous analysis, prior results generally focusing on variants in which noise is introduced into the dynamics, the resulting system being amenable to standard techniques from statistical mechanics or stochastic evolutionary game theory [25]. A series of recent papers [6,3,4], has seen the first rigorous analyses of 1-dimensional unperturbed Schelling models, in an asymptotic framework largely unknown in statistical mechanics. Here we provide the first such analysis of 2-and 3-dimensional unperturbed models, establishing most of the phase diagram, and answering a challenge from [6].

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Section: Introductionmentioning

confidence: 86%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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Unperturbed Schelling Segregation in Two or Three Dimensions

2016

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“…They prove that the expected diameter of the segregated region containing the origin in the final configuration grows at least exponentially in w 2 . See also [BEL14,BEL15c,BEL15a,BEL15b] for recent rigorous results on the unperturbed Schelling model in one, two, and three dimensions. The models studied in these papers have a more general initial configuration of types and more general tolerance parameters than the models in [BIKK12,IKLZ15] and the current paper.…”

Section: Historymentioning

confidence: 99%

Scaling limits of the Schelling model

Holden

Sheffield⋆

2019

Probab. Theory Relat. Fields

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The Schelling model, 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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Self-organized Segregation on the Grid

Omidvar¹,

Franceschetti²

2017

J Stat Phys

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We consider an agent-based model in which two types of agents interact locally over a graph and have a common intolerance threshold $\tau$ for changing their types with exponentially distributed waiting times. The model is equivalent to an unperturbed Schelling model of self-organized segregation, an Asynchronous Cellular Automata (ACA) with extended Moore neighborhoods, or a zero-temperature Ising model with Glauber dynamics, and has applications in the analysis of social and biological networks, and spin glasses systems. Some rigorous results were recently obtained in the theoretical computer science literature, and this work provides several extensions. We enlarge the intolerance interval leading to the formation of large segregated regions of agents of a single type from the known size $\epsilon>0$ to size $\approx 0.134$. Namely, we show that for $0.433 < \tau < 1/2$ (and by symmetry $1/2<\tau<0.567$), the expected size of the largest segregated region containing an arbitrary agent is exponential in the size of the neighborhood. We further extend the interval leading to large segregated regions to size $\approx 0.312$ considering "almost segregated" regions, namely regions where the ratio of the number of agents of one type and the number of agents of the other type vanishes quickly as the size of the neighborhood grows. In this case, we show that for $0.344 < \tau \leq 0.433$ (and by symmetry for $0.567 \leq \tau<0.656$) the expected size of the largest almost segregated region containing an arbitrary agent is exponential in the size of the neighborhood. The exponential bounds that we provide also imply that complete segregation, where agents of a single type cover the whole grid, does not occur with high probability for $p=1/2$ and the range of tolerance considered

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Tipping Points in 1-Dimensional Schelling Models with Switching Agents

Cited by 15 publications

References 24 publications

Unperturbed Schelling Segregation in Two or Three Dimensions

Unperturbed Schelling Segregation in Two or Three Dimensions

Scaling limits of the Schelling model

Self-organized Segregation on the Grid

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