Title: Emergence of near-TAP free energy functional in the SK model at high temperature Show Chinese title

Title: SK模型在高温下接近TAP自由能泛函的出现

Abstract: We study the SK model at inverse temperature $\beta>0$ and strictly positive field $h>0$ in the region of $(\beta,h)$ where the replica-symmetric formula is valid. An integral representation of the partition function derived from the Hubbard-Stratonovitch transformation combined with a duality formula is used to prove that the infinite volume free energy of the SK model can be expressed as a variational formula on the space of magnetisations, $m$. The resulting free energy functional differs from that of Thouless, Anderson and Palmer (TAP) by the term $-\frac{\beta^2}{4}\left(q-q_{\text{EA}}(m)\right)^2$ where $q_{\text{EA}}(m)$ is the Edwards-Anderson parameter and $q$ is the minimiser of the replica-symmetric formula. Thus, both functionals have the same critical points and take the same value on the subspace of magnetisations satisfying $q_{\text{EA}}(m)=q$. This result is based on an in-depth study of the global maximum of this near-TAP free energy functional using Bolthausen's solutions of the TAP equations, Bandeira & van Handel's bounds on the spectral norm of non-homogeneous Wigner-type random matrices, and Gaussian comparison techniques. It holds for $(\beta,h)$ in a large subregion of the de Almeida and Thouless high-temperature stability region. Show Chinese abstract

Abstract: 我们研究了 SK 模型在倒温度为$\beta>0$和严格正场强$h>0$的情况下，且在复制对称公式成立的区域$(\beta,h)$中。通过结合 Hubbard-Stratonovitch 变换和对偶公式得出的配分函数积分表示，证明了 SK 模型的无穷体积自由能可以在磁化强度空间$m$上表示为变分公式。由此得到的自由能泛函与 Thouless、Anderson 和 Palmer (TAP) 的自由能泛函相差一项$-\frac{\beta^2}{4}\left(q-q_{\text{EA}}(m)\right)^2$，其中$q_{\text{EA}}(m)$是 Edwards-Anderson 参数，$q$是复制对称公式最小值的极小化子。 因此，这两个泛函具有相同的临界点，并且在满足 $q_{\text{EA}}(m)=q$的磁化子空间上取相同的值。 该结果基于对近似TAP自由能泛函的全局最大值的深入研究，使用了Bolthausen提出的TAP方程解、Bandeira与van Handel关于非齐次Wigner型随机矩阵谱范数的界以及高斯比较技术。 它对于de Almeida和Thouless高温稳定性区域的一个大子区域内任意$(\beta,h)$成立。