Ising model on trees and factors of IID

2020/12/17 by Nam, Danny, Sly, Allan, Zhang, Lingfu
#37A35 #37A50 #60G10 #60K35 #Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR)

paper · doi:10.48550/arxiv.2012.09484

Abstract

We study the ferromagnetic Ising model on the infinite d-regular tree under the free boundary condition. This model is known to be a factor of IID in the uniqueness regime, when the inverse temperature β≥ 0 satisfies \tanh β≤ (d-1)^-1. However, in the reconstruction regime (\tanh β> (d-1)^-(1)/(2)), it is not a factor of IID. We construct a factor of IID for the Ising model beyond the uniqueness regime via a strong solution to an infinite dimensional stochastic differential equation which partially answers a question of Lyons. The solution \X_t(v) \ of the SDE is distributed as X_t(v) = tτ_v + B_t(v), where \τ_v \ is an Ising sample and \B_t(v) \ are independent Brownian motions indexed by the vertices in the tree. Our construction holds whenever \tanh β≤ c(d-1)^-(1)/(2), where c>0 is an absolute constant.

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