Ising和Fuzzy Potts模型的随机控制
Stochastic domination for the Ising and fuzzy Potts models
摘要：
我们讨论了伊辛模型和模糊Potts模型的随机控制的各个方面。我们首先考虑度$d$，$\Td$的同构树上的伊辛模型。对于给定的相互作用参数$J_1$，$J_2>0$和外场$h_1\in\RR$，我们计算最小外场$\tilde{h}$，使得对于所有$h\geq\tilde{h}$，参数$J_2$和$h$的正测度优于参数$J_1$和$h_1$的正测度.讨论了$\tilde{h}$关于三个参数$J_1$，$J_2$，$h$的连续性，以及在固定外场下加测度在相互作用参数中的随机序.其次，我们考虑模糊Potts模型，并证明了在$\Zd$上模糊Potts测度支配同一组乘积测度，而在$\Td$上，对于一定的参数值，自由和负模糊Potts测度支配不同的乘积测度。对于伊辛模型，Liggett和Steif证明了在$\Zd$上，正测度支配同一组乘积测度，而在$\T^2$上，除非有唯一的相位，否则这种说法完全失败。
英文摘要：
We discuss various aspects concerning stochastic domination for the Ising model and the fuzzy Potts model. We begin by considering the Ising model on the homogeneous tree of degree $d$, $\Td$. For given interaction parameters $J_1$, $J_2>0$ and external field $h_1\in\RR$, we compute the smallest external field $\tilde{h}$ such that the plus measure with parameters $J_2$ and $h$ dominates the plus measure with parameters $J_1$ and $h_1$ for all $h\geq\tilde{h}$. Moreover, we discuss continuity of $\tilde{h}$ with respect to the three parameters $J_1$, $J_2$, $h$ and also how the plus measures are stochastically ordered in the interaction parameter for a fixed external field. Next, we consider the fuzzy Potts model and prove that on $\Zd$ the fuzzy Potts measures dominate the same set of product measures while on $\Td$, for certain parameter values, the free and minus fuzzy Potts measures dominate different product measures. For the Ising model, Liggett and Steif proved that on $\Zd$ the plus measures dominate the same set of product measures while on $\T^2$ that statement fails completely except when there is a unique phase.
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