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

  • C. Douglas Howard (a1)


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 θ c 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.


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Postal address: Baruch College, Box G0930, 17 Lexington Avenue, New York, NY 10010, USA. Email address:


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



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[4] Howard, C. D. In preparation.
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