论文标题
Nishimori线上的Gibbs-Bogoliubov不平等
Gibbs-Bogoliubov inequality on Nishimori line
论文作者
论文摘要
Gibbs-Bogoliubov不平等指出，系统的自由能总是低于试验函数计算的自由能。在这项研究中，我们表明，Gibbs-Bogoliubov不平等的对应物在Nishimori线上，用于具有高斯随机性的旋转玻璃模型。我们的不平等指出，系统的淬灭自由能总是低于使用淬火试验功能计算的。证明的关键组成部分是压力函数的凸度$ \ mathbb {e} \ left [\ log log z_ {} \ right] $相对于沿Nishimori线的参数，这与相对于逆温度的常规凸度有所不同。当我们的不平等应用于均值场模型（例如Sherrington-Kirkpatrick模型和$ P $ -SPIN模型）时，界限与副本对称解的结合表明等于相等。
The Gibbs-Bogoliubov inequality states that the free energy of a system is always lower than that calculated by a trial function. In this study, we show that a counterpart of the Gibbs-Bogoliubov inequality holds on the Nishimori line for Ising spin-glass models with Gaussian randomness. Our inequality states that the quenched free energy of a system is always lower than that calculated using a quenched trial function. The key component of the proof is the convexity of the pressure function $\mathbb{E}\left[\log Z_{} \right]$ with respect to the parameters along the Nishimori line, which differs from the conventional convexity with respect to the inverse temperature. When our inequality was applied to mean-field models, such as the Sherrington-Kirkpatrick model and $p$-spin model, the bound coincided with the replica-symmetric solution indicating that the equality holds.
