树上的临界Ising模型、凹递归和非线性容量
The critical Ising model on trees, concave recursions and nonlinear capacity
摘要：
我们认为伊辛模型的一般树在各种边界条件下：所有加，自由和自旋玻璃。在每一种情况下，我们确定当边界向无穷远退时，根何时受到极限中边界值的影响。我们得到确切的能力标准，在临界温度下的行为。对于正边界条件，出现$L^3$容量。特别是，在一个球对称树，$n^αb^n$顶点在$n$级别（有界因子），我们证明了有一个唯一的吉布斯措施的铁磁伊辛模型在相关的临界温度当且仅当$α\le1/2$。我们的证明是基于树和$L^p$能力的非线性递归之间的一个新的联系。
英文摘要：
We consider the Ising model on a general tree under various boundary conditions: all plus, free and spin-glass. In each case, we determine when the root is influenced by the boundary values in the limit as the boundary recedes to infinity. We obtain exact capacity criteria that govern behavior at critical temperatures. For plus boundary conditions, an $L^3$ capacity arises. In particular, on a spherically symmetric tree that has $n^αb^n$ vertices at level $n$ (up to bounded factors), we prove that there is a unique Gibbs measure for the ferromagnetic Ising model at the relevant critical temperature if and only if $α\le1/2$. Our proofs are based on a new link between nonlinear recursions on trees and $L^p$ capacities.
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