论文标题
关于SK模型中踢踏自由能的凹入性
On concavity of TAP free energy in the SK model
论文作者
论文摘要
我们分析了Sherrington-kirkpatrick模型的thouless-Anderson-Palmer（Tap）自由能的Hessian，在Bolthausen的Tap方程的大概解决方案中评估了De Almeida-thou thouless Line。我们表明，经验光谱分布薄弱地收敛到一个低于AT线的负支撑的度量，其支撑在AT线上包括零。从这种``宏观''意义上说，踢踏自由能在理论的顺序参数（即随机自旋磁化）中凹入。这证明了对AT线的光谱解释。但是，对于特定的磁化，自来水能量的Hessian可以具有正分离的特征值。是否可能也可能发生此类离群值靠近TAP解决方案的问题。在简化的环境中，磁性独立于该疾病，我们证明了Plefka的第二条件等于所有特征值均为负。
We analyse the Hessian of the Thouless-Anderson-Palmer (TAP) free energy for the Sherrington-Kirkpatrick model, below the de Almeida-Thouless line, evaluated in Bolthausen's approximate solutions of the TAP equations. We show that the empirical spectral distribution weakly converges to a measure with negative support below the AT line, an that the support includes zero on the AT line. In this ``macroscopic'' sense, TAP free energy is concave in the order parameter of the theory, i.e. the random spin-magnetisations. This proves a spectral interpretation of the AT line. However, for specific magnetizations, the Hessian of the TAP free energy can have positive outlier eigenvalues. The question whether such outliers may also occur close to the TAP solutions is left open. In a simplified setting where the magnetizations are independent of the disorder, we prove that Plefka's second condition is equivalent to all eigenvalues being negative.
