论文标题
在树木和IID因素上的模型
Ising model on trees and factors of IID
论文作者
论文摘要
我们在自由边界条件下研究了无限$ d $ regratular树上的铁磁伊辛模型。已知该模型在唯一性方面是IID的一个因素，当反向温度$β\ ge 0 $满足$ \ tanhβ\ le（d-1）^{ - 1} $时。但是，在重建体制（$ \tanhβ>（d-1）^{ - \ frac {1} {2}} $）中，这不是IID的因素。我们通过对无限维度随机微分方程的强大解决方案为ISING模型构建了IID因素，该方程部分回答了里昂的问题。 SDE的解决方案$ \ {x_t（v）\} $被分发为 \ [ x_t（v）=tτ_v + b_t（v）， \]其中$ \ {τ_v\} $是一个样本，$ \ {b_t（v）\} $是树上的顶点索引的独立布朗动议。每当$ \tanhβ\ le c（d-1）^{ - \ frac {1} {2}} $时，我们的构造就会成立，其中$ c> 0 $是绝对常数。
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 $β\ge 0$ satisfies $\tanh β\le (d-1)^{-1}$. However, in the reconstruction regime ($\tanh β> (d-1)^{-\frac{1}{2}}$), 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 $\{τ_v \}$ is an Ising sample and $\{B_t(v) \}$ are independent Brownian motions indexed by the vertices in the tree. Our construction holds whenever $\tanh β\le c(d-1)^{-\frac{1}{2}}$, where $c>0$ is an absolute constant.
