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Zero-temperature ising spin dynamics on the homogeneous tree of degree three

Part of: Special processes Time-dependent statistical mechanics (dynamic and nonequilibrium)

Published online by Cambridge University Press:  14 July 2016

C. Douglas Howard
C. Douglas Howard*
Affiliation:
Baruch College
*
∗Postal address: Baruch College, Box G0930, 17 Lexington Avenue, New York, NY 10010, USA. Email address: dhoward@baruch.cuny.edu
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Abstract

We investigate zero-temperature dynamics for a homogeneous ferromagnetic Ising model on the homogeneous tree of degree three (𝕋) with random (i.i.d. Bernoulli) spin configuration at time 0. Letting θ denote the probability that any particular vertex has a +1 initial spin, for infinite spin clusters do not exist at time 0 but we show that infinite ‘spin chains’ (doubly infinite paths of vertices with a common spin) exist in abundance at any time ϵ > 0. We study the structure of the subgraph of 𝕋 generated by the vertices in time-ϵ spin chains. We show the existence of a phase transition in the sense that, for some critical θc with spin chains almost surely never form for θ < θc but almost surely do form in finite time for θ > θc. We relate these results to certain quantities of physical interest, such as the t → ∞ asymptotics of the probability that any particular vertex changes spin after time t.

Keywords

Glauber dynamicsIsing spin dynamics

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory 82C44: Dynamics of disordered systems (random Ising systems, etc.)
Secondary: 82C26: Dynamic and nonequilibrium phase transitions (general)
Type
Research Papers
Information
Journal of Applied Probability , Volume 37 , Issue 3 , September 2000 , pp. 736 - 747
Copyright
Copyright Š Applied Probability Trust 2000 

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Footnotes

Research supported in part by NSF Grant DMS-98-15226.

References

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