Phase ordering after a deep quench: the stochastic Ising and hard core gas models on a tree

Caputo, Pietro; Martinelli, Fabio

doi:10.1007/s00440-005-0475-y

Cited by 20 publications

(17 citation statements)

References 33 publications

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“…Now using Azuma's inequality, there exist constants c 1 , c 2 ∈ R + such that It follows by an application of Boole's inequality that there exists a constant c ∈ R + such that (29) holds.…”

Section: Appendixmentioning

confidence: 99%

Self-organized Segregation on the Grid

Omidvar

Franceschetti

2017

Proceedings of the ACM Symposium on Principles of Distributed Computing

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We consider an agent-based model with exponentially distributed waiting times in which two types of agents interact locally over a graph, and based on this interaction and on the value of a common intolerance threshold τ , decide whether to change their types. This is equivalent to a zero-temperature Ising model with Glauber dynamics, an Asynchronous Cellular Automaton (ACA) with extended Moore neighborhoods, or a Schelling model of self-organized segregation in an open system, 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 expected formation of large segregated regions of agents of a single type from the known size > 0 to size ≈ 0.134. Namely, we show that for 0.433 < τ < 1/2 (and by symmetry 1/2 < τ < 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 expected large segregated regions to size ≈ 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 < τ ≤ 0.433 (and by symmetry for 0.567 ≤ τ < 0.656) the expected size of the largest almost segregated region containing an arbitrary agent is exponential in the size of the neighborhood. This behavior is reminiscent of supercritical percolation, where small clusters of empty sites can be observed within any sufficiently large region of the occupied percolation cluster. 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 intolerance considered.

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Section: Appendixmentioning

confidence: 99%

Self-organized Segregation on the Grid

Omidvar

Franceschetti

2017

Proceedings of the ACM Symposium on Principles of Distributed Computing

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“…Morris [28] has shown that p * converges to 1/2 as d → ∞. Caputo and Martinelli [29] have shown the same result for d-regular trees, while Kanoria and Montanari [30] derived it for d-regular trees in a synchronous setting where flips occur simultaneously, and obtained lower bounds on p * (d) for small values of d. The case d = 1 was first investigated by Erdös and Ney [31], and Arratia [32] has proven that p * (1) = 1.…”

Section: B Contributionmentioning

confidence: 98%

“…It is also easy to see that M 0 = E[W ] = N/2, and M N = W . We also have It follows by an application of Boole's inequality that there exists a constant c ∈ R + such that (29) holds.…”

Section: Appendixmentioning

confidence: 99%

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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“…Indeed, Howard [21] showed that p c (T 3 ) > 1/2, and it was proved by Caputo and Martinelli [13] that p c (T d ) → 1/2 as d → ∞ (in fact their result is more general, and this statement is straightforward to prove in the zero-temperature case), but for every d ≥ 4 it is unknown whether or not p c (T d ) = 1/2. For further results and problems about the case p = 1/2, on Z d and on other graphs, see for example [12,21,29,32,33]; for a good account of Glauber dynamics at non-zero temperatures, see [25].…”

Section: Introductionmentioning

confidence: 94%

Zero-temperature Glauber dynamics on $${\mathbb{Z}^d}$$

Morris

2009

Probab. Theory Relat. Fields

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We study zero-temperature Glauber dynamics on Z d , which is a dynamic version of the Ising model of ferromagnetism. Spins are initially chosen according to a Bernoulli distribution with density p, and then the states are continuously (and randomly) updated according to the majority rule. This corresponds to the sudden quenching of a ferromagnetic system at high temperature with an external field, to one at zero temperature with no external field. Define p c (Z d ) to be the infimum over p such that the system fixates at '+' with probability 1. It is a folklore conjecture that p c (Z d ) = 1/2 for every 2 ≤ d ∈ N. We prove that p c (Z d ) → 1/2 as d → ∞.

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Phase ordering after a deep quench: the stochastic Ising and hard core gas models on a tree

Cited by 20 publications

References 33 publications

Self-organized Segregation on the Grid

Self-organized Segregation on the Grid

Self-organized Segregation on the Grid

Zero-temperature Glauber dynamics on $${\mathbb{Z}^d}$$

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