标题： 弱相关性范围内富集连续随机能量模型的自由能 显示英文标题

标题： The Free Energy of an Enriched Continuous Random Energy Model in the Weak Correlation Regime

摘要： 我们使用平均场无序系统的Hamilton-Jacobi方法重新审视连续随机能量模型（CREM）极限自由能的证明。 为此，我们引入了一个增强模型，该模型在CREM和Ruelle概率级联之间进行插值。 我们关注弱相关性区域，在此区域内，CREM的协方差函数$A$被恒等函数所上限。 在弱相关性区域，我们证明自由能由Hopf公式给出。 所得表达式与$A$无关，证实在此区域内自由能不依赖于协方差函数的具体形式。 在弱相关性区域之外，Hamilton-Jacobi框架不再适用。 此外，我们提供了一个例子，说明对相关变分原理的正式应用未能得出正确的自由能。 显示英文摘要

摘要： We revisit the proof of the limiting free energy of the continuous random energy model (CREM) using the Hamilton--Jacobi approach for mean-field disordered systems. To achieve this, we introduce an enriched model that interpolates between the CREM and the Ruelle probability cascade. We focus on the weak correlation regime, where the CREM's covariance function $A$ is bounded above by the identity function. In the weak correlation regime, we show that the free energy is given by the Hopf formula. The resulting expression is independent of $A$, confirming that in this regime the free energy does not depend on the precise form of the covariance function. Outside of the weak correlation regime, the Hamilton--Jacobi framework no longer applies. Moreover, we provide an example where a formal application of the associated variational principle fails to yield the correct free energy.