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. The solution $\{X_t(v)\}$ of the SDE is distributed as \[ X_t(v) = tτ_v + B_t(v), \] 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.

34 pages. CMP to appear

References in corpus (3)

• Local algorithms for independent sets are half-optimal
• Proper 3-colorings of ${\mathbb{Z}}^2$ are Bernoulli
• Stationary Distributions for the Voter Model in $d\ge 3$ are Factors of IID