广义随机能量模型的变分界
Variational Bounds for the Generalized Random Energy Model
摘要：
通过建立变分上下界，计算了随机能量模型（REM）和广义随机能量模型（GREM）的压力。对于上界，我们推广了Guerra的“破副本对称界”，并确定随机概率级联作为模型的适当随机重叠结构。对于REM的下限得到，在高温区使用Talagrand的浓度测量不等式，并在低温区使用凸性和高温公式。GREM的下限是通过归纳从REM的下限得出的。虽然对下界的论证是相当标准的，但我们对上界的证明是新的。
英文摘要：
We compute the pressure of the random energy model (REM) and generalized random energy model(GREM) by establishing variational upper and lower bounds. For the upper bound, we generalize Guerra's ``broken replica symmetry bounds",and identify the random probability cascade as the appropriate random overlap structure for the model. For the REM the lower bound is obtained, in the high temperature regime using Talagrand's concentration of measure inequality, and in the low temperature regime using convexity and the high temperature formula. The lower bound for the GREM follows from the lower bound for the REM by induction. While the argument for the lower bound is fairly standard, our proof of the upper bound is new.
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