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

  • C. Douglas Howard (a1)

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 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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Corresponding author

Postal address: Baruch College, Box G0930, 17 Lexington Avenue, New York, NY 10010, USA. Email address: dhoward@baruch.cuny.edu

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

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References

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[1] Derrida, B. (1995). Exponents appearing in the zero-temperature dynamics of the 1D Potts model. J. Phys. A. 28, 14811491.
[2] Derrida, B., Hakim, V., and Pasquier, V. (1995). Exact first-passage exponents of 1D domain growth: relation to a reaction-diffusion model. Phys. Rev. Lett. 75, 751754.
[3] Derrida, B., de Oliveira, P. M. C., and Stauffer, D. (1996). Stable spins in the zero temperature spinodal decomposition of 2D Potts models. Physica 224A, 604612.
[4] Howard, C. D. In preparation.
[5] Lyons, R. Private communication.
[6] Nanda, S., Newman, C. M., and Stein, D. L. (2000). To appear in Dynamics of Ising spin systems at zero temperature. On Dobrushin's Way (from Probability Theory to Statistical Physics). Eds. Minlos, R., Shlosman, S. and Suhov, Y. American Mathematical Society, Providence.
[7] Newman, C. M., and Stein, D. L. (1999). Blocking and persistence in the zero-temperature dynamics of homogeneous and disordered Ising models. Phys. Rev. Lett. 82, 39443947.
[8] Stauffer, D. (1994). Ising spinodal decomposition at T = 0 in one to five dimensions. J. Phys. A 27, 50295032.

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