Ising Model on Trees and Factors of IID

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 (tanhβ>(d-1)-12), 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 (Comb Probab Comput 2(2):285–300, 2017). The solution { X_t(v) } of the SDE is distributed as Xt(v)=tτv+Bt(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 tanhβ≤c(d-1)-12, where c> 0 is an absolute constant.