Zero-temperature dynamics of the weakly disordered Ising model

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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Zero-temperature dynamics for the ferromagnetic Ising model on random graphs

Physica A Statistical Mechanics and its Applications ◽

10.1016/s0378-4371(02)00797-5 ◽

2002 ◽

Vol 310 (3-4) ◽

pp. 275-284 ◽

Cited By ~ 25

Author(s):

Olle Häggström

Keyword(s):

Ising Model ◽

Random Graphs ◽

Zero Temperature ◽

Temperature Dynamics ◽

Ferromagnetic Ising Model

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

Journal of Applied Probability ◽

10.1239/jap/1014842832 ◽

2000 ◽

Vol 37 (3) ◽

pp. 736-747 ◽

Cited By ~ 11

Author(s):

C. Douglas Howard

Keyword(s):

Phase Transition ◽

Ising Model ◽

Spin Dynamics ◽

Spin Chains ◽

Homogeneous Tree ◽

Zero Temperature ◽

Spin Configuration ◽

Ising Spin ◽

Temperature Dynamics ◽

Ferromagnetic Ising Model

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.

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