Ising Model on Trees and Factors of IID

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 $$\beta \ge 0$$ satisfies $$\tanh \beta \le (d-1)^{-1}$$ . However, in the reconstruction regime ( $$\tanh \beta > (d-1)^{-\frac{1}{2}}$$ ), 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 (Comb Probab Comput 2(2):285–300, 2017). The solution $$\{X_t(v) \}$$ of the SDE is distributed as $$\begin{aligned} X_t(v) = t\tau _v + B_t(v), \end{aligned}$$ 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.