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Improved Intolerance Intervals and Size Bounds for a Schelling-Type Spin System
Social and Information Networks · Submitted 24 Oct 2018
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim This paper extends the above result to the interval $\tau \in (\sim 0.433,\sim 0.567) \setminus \{1/2\}$.
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The abstract, sentence by sentence
We consider a Schelling model of self-organized segregation in an open system that is equivalent to a zero-temperature Ising model with Glauber dynamics, or an Asynchronous Cellular Automaton (ACA) with extended Moore neighborhoods. Previous work has shown that if the intolerance parameter of the model $\tau\in (\sim 0.488,\sim 0.512) \setminus \{1/2\}$, then for a sufficiently large neighborhood of interaction $N$, any particle will end up in an exponentially large monochromatic region almost surely. This paper extends the above result to the interval $\tau \in (\sim 0.433,\sim 0.567) \setminus \{1/2\}$. We also improve the bounds on the size of the monochromatic region by exponential factors in $N$. Finally, we show that when particles are placed on the infinite lattice $\mathbb{Z}^2$ rather than on a flat torus, for the values of $\tau$ mentioned above, sufficiently large $N$, and after a sufficiently long evolution time, any particle is contained in a large monochromatic region of size exponential in $N$, almost surely. The new proof, critically relies on a novel geometric construction related to the formation of the monochromatic region.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We consider a Schelling model of self-organized segregation in an open system that is equivalent to a zero-temperature Ising model with Glauber dynamics, or an Asynchronous Cellular Automaton (ACA) with extended Moore neighborhoods. | background 0.72 | 0.21 | 0.03 | asserted 1.00 |
| s1 | Previous work has shown that if the intolerance parameter of the model $\tau\in (\sim 0.488,\sim 0.512) \setminus \{1/2\}$, then for a sufficiently large neighborhood of interaction $N$, any particle will end up in an exponentially large monochromatic region almost surely. | background 1.00 | 0.03 | 0.05 | asserted 1.00 |
| s2 | This paper extends the above result to the interval $\tau \in (\sim 0.433,\sim 0.567) \setminus \{1/2\}$. | result 0.52 | 0.94 | 0.68 | asserted 0.99 |
| s3 | We also improve the bounds on the size of the monochromatic region by exponential factors in $N$. | result 0.66 | 0.95 | 0.11 | asserted 1.00 |
| s4 | Finally, we show that when particles are placed on the infinite lattice $\mathbb{Z}^2$ rather than on a flat torus, for the values of $\tau$ mentioned above, sufficiently large $N$, and after a sufficiently long evolution time, any particle is contained in a large monochromatic region of size exponential in $N$, almost surely. | result 0.95 | 0.94 | 0.29 | asserted 1.00 |
| s5 | The new proof, critically relies on a novel geometric construction related to the formation of the monochromatic region. | method 0.92 | 0.90 | 0.04 | asserted 1.00 |
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Original abstract
We consider a Schelling model of self-organized segregation in an open system that is equivalent to a zero-temperature Ising model with Glauber dynamics, or an Asynchronous Cellular Automaton (ACA) with extended Moore neighborhoods. Previous work has shown that if the intolerance parameter of the model $\tau\in (\sim 0.488,\sim 0.512) \setminus \{1/2\}$, then for a sufficiently large neighborhood of interaction $N$, any particle will end up in an exponentially large monochromatic region almost surely. This paper extends the above result to the interval $\tau \in (\sim 0.433,\sim 0.567) \setminus \{1/2\}$. We also improve the bounds on the size of the monochromatic region by exponential factors in $N$. Finally, we show that when particles are placed on the infinite lattice $\mathbb{Z}^2$ rather than on a flat torus, for the values of $\tau$ mentioned above, sufficiently large $N$, and after a sufficiently long evolution time, any particle is contained in a large monochromatic region of size exponential in $N$, almost surely. The new proof, critically relies on a novel geometric construction related to the formation of the monochromatic region.
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