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 $\beta \ge 0$ satisfies $\tanh \beta \le (d-1)^{-1}$\mathrm{.}However,inthereconstructionregime($\tanh \beta > (d-1)^{-\frac{1}{2}}$), 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 $$X_t(v)=t{\tau }_v+B_t(v),$$ where $\{{\tau }_v\}$ is an Ising sample and $\{B_t(v)\}$ are independent Brownian motions indexed by the vertices in the tree. Our construction holds whenever $\tanh \beta \le c(d-1{)}^{-\frac{1}{2}}$, where $c>0$ is an absolute constant.