多维自旋SK模型的Aizenman-Sims-Starr和Guerra格式
The Aizenman-Sims-Starr and Guerra's schemes for the SK model with multidimensional spins
摘要：
我们证明了多维的Sherrington-Kirkpatrick模型中自由能的上界和下界（例如，Heisenberg）自旋的方法，得到了基于相应Parisi泛函的变分不等式.我们采用比较方案的Aizenman，西姆斯和斯塔尔和一个涉及广义随机能量模型启发的过程和Ruelle的概率级联的Guerra。为此目的，得到了Gaertner-Ellis型的抽象淬灭大偏差原理。利用Ruelle的概率级联和Bolthausen-Sznitman结合的性质，我们推导出我们模型的Guerra余项的Talagrand表示。通过与一类非线性偏微分方程建立联系，研究了多维Parisi泛函的性质。解决Talagrand提出的一个问题，我们证明了严格凸的局部Parisi功能。我们证明了Parisi公式的情况下，使用Talagrand的先验估计的方法的自旋的多维高斯先验分布的本地自由能。
英文摘要：
We prove upper and lower bounds on the free energy in the Sherrington-Kirkpatrick model with multidimensional (e.g., Heisenberg) spins in terms of the variational inequalities based on the corresponding Parisi functional. We employ the comparison scheme of Aizenman, Sims and Starr and the one of Guerra involving the generalised random energy model-inspired processes and Ruelle's probability cascades. For this purpose an abstract quenched large deviations principle of the Gaertner-Ellis type is obtained. Using the properties of Ruelle's probability cascades and the Bolthausen-Sznitman coalescent, we derive Talagrand's representation of the Guerra remainder term for our model. We study the properties of the multidimensional Parisi functional by establishing a link with a certain class of the non-linear partial differential equations. Solving a problem posed by Talagrand, we show the strict convexity of the local Parisi functional. We prove the Parisi formula for the local free energy in the case of the multidimensional Gaussian a priori distribution of spins using Talagrand's methodology of the a priori estimates.
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