Universal structures in some mean field spin glasses, and an application

谱独立性的普适性及其在自旋玻璃快速混合中的应用

Universality of Spectral Independence with Applications to Fast Mixing in Spin Glasses

该研究证明谱独立性具有普适性，即O(n)弛豫时间可推出谱独立性；还证明哈密顿量多线性延拓光滑的分布的Glauber动力学有O(n)弛豫时间，且这类分布分数对数凹、满足修正对数Sobolev不等式，还得到高温混合p-spin模型的最优O~(n)混合时间，对多领域有重要意义。

关于平均场自旋玻璃中副本对称性的不连续破缺与碎裂

On the Discontinuous Breaking of Replica Symmetry and Shattering in Mean-Field Spin Glasses

本文证明平均场自旋玻璃中副本对称性在 $β_c$ 处不连续破缺可推出中间碎裂相的存在，并给出一个球形自旋玻璃反例，说明碎裂现象并不必然伴随不连续相变。

Existence of the free energy for heavy-tailed spin glasses

We study the free energy of a mean-field spin glass whose coupling distribution has power law tails. Under the assumption that the couplings have infinite variance and finite mean, we show that the thermodynamic limit of the quenched free energy exists, and that the free energy is self-averaging.

High Temperature Asymptotics of Orthogonal Mean-Field Spin Glasses

We evaluate the high temperature limit of the free energy of spin glasses on the hypercube with Hamiltonian $H_N(σ) = σ^T J σ$, where the coupling matrix $J$ is drawn from certain symmetric orthogonally invariant ensembles. Our derivation relates the annealed free energy of these models to a spherical integral, and expresses the limit of the free energy in terms of the limiting spectral measure of the coupling matrix $J$. As an application, we derive the limiting free energy of the Random Orthogonal Model (ROM) at high temperatures, which confirms non-rigorous calculations of Marinari et al. (…

自由能的严格普遍性二元性：球形铁磁体模型

A Sharp Universality Dichotomy for the Free Energy of Spherical Spin Glasses

研究球形铁磁体模型自由能的严格普遍性二元性，通过不同尾部条件下的变分公式和极值统计分析，揭示了自由能行为的统一分类。

平均场自旋玻璃中的局部独立性

Local independence in mean-field spin glasses

本文提出多空腔位点的严格空腔方法，证明平均场自旋玻璃中局部独立性与薄壳、重叠集中等复本对称条件等价，并统一涵盖 SK 与 Gardner 模型。

优化带外场的平均场自旋玻璃

Optimizing Mean Field Spin Glasses with External Field

针对带非平凡外场的平均场自旋玻璃，论文提出满足无重叠间隙条件时的两阶段消息传递近似最大化算法，并给出可构造完整超度量极大值树的分支变体。

Spectral gap estimates in mean field spin glasses

We show that mixing for local, reversible dynamics of mean field spin glasses is exponentially slow in the low temperature regime. We introduce a notion of free energy barriers for the overlap, and prove that their existence imply that the spectral gap is exponentially small, and thus that mixing is exponentially slow. We then exhibit sufficient conditions on the equilibrium Gibbs measure which guarantee the existence of these barriers, using the notion of replicon eigenvalue and 2D Guerra Talagrand bounds. We show how these sufficient conditions cover large classes of Ising spin models for re…

High temperature TAP upper bound for the free energy of mean field spin glasses

This work proves an upper bound for the free energy of the Sherrington-Kirkpatrick model and its generalizations in terms of the Thouless-Anderson-Palmer (TAP) energy. The result applies to models with spherical or Ising spins and any mixed $p$-spin Hamiltonian with external field or with a non-linear spike term. The bound is expected to be tight to leading order at high temperature, and is non-trivial in the presence of an external field. For the proof a geometric microcanonical method is employed, in which one covers the spin space with sets, each of which is centered at a magnetization vect…

随机图与自旋玻璃中的最大稳定局部最优：相变与普适性

Maximally-stable Local Optima in Random Graphs and Spin Glasses: Phase Transitions and Universality

本研究证明随机图与自旋玻璃中h-稳定局部最优存在普适相变，证实Behrens猜想，并将友好划分猜想推广至一般h值。

关于无序系统的零点与算法：平均场自旋玻璃

On zeros and algorithms for disordered systems: mean-field spin glasses

本文针对平均场自旋玻璃提出确定性拟多项式时间算法，通过研究配分函数零点位置，在第二矩机制下以任意精度估计配分函数，并成功覆盖Sherrington–Kirkpatrick模型的整个复制对称相。