AI 中文总结
本文研究临界逆温度β=1下两类Sherrington-Kirkpatrick模型的双副本重叠分布，通过球-立方体比较原理等方法确定其极限分布，回答了Talagrand的问题，相关论证由GPT-5.6 Pro生成。
AI 中文摘要
我们研究了Ising和球形Sherrington-Kirkpatrick模型在临界逆温度β=1下的双副本重叠R_{1,2}的分布。主要结果表明，在两种模型中，R_{1,2}的尺度为N^{-1/3}，且N^{1/3}R_{1,2}的淬火分布收敛于由反射Airy₁点过程定义的显式随机概率测度。由此，我们确定了N^{2/3}E⟨R_{1,2}²⟩的极限值，回答了Talagrand提出的问题[1]。对于球形SK模型，我们通过将吉布斯测度表示为ℝᴺ上各向异性高斯分布（满足范数为√N的条件），再过渡到GOE谱边缘的Airy₁标度极限来得到该极限。对于Ising SK模型，证明基于球-立方体比较原理，表明球形和Ising吉布斯测度下N^{1/3}R_{1,2}的淬火分布渐近重合。本文是文献[2]的姊妹篇，在该文献中我们引入了相关比较原理以确定SK自由能的极限涨落。本文的大部分论证由GPT-5.6 Pro生成，目的是探索该工作所发展思想的进一步推论。
英文摘要
We study the distribution of the two-replica overlap $R_{1,2}$ in the Ising and spherical Sherrington-Kirkpatrick models at the critical inverse temperature $β= 1$. Our main result shows that in both models, $R_{1,2}$ has scale $N^{-1/3}$, and the quenched distribution of $N^{1/3} R_{1,2}$ converges to an explicit random probability measure defined in terms of the reflected $\mathrm{Airy}_1$ point process. As a consequence, we characterize the limiting value of $N^{2/3} \mathbb{E} \langle R_{1,2}^2 \rangle$, answering a question of Talagrand \cite{talagrand2011mean2}. For the spherical SK model, we obtain the limit by representing the Gibbs measure as an anisotropic Gaussian on $\mathbb{R}^N$ conditioned to have norm $\sqrt{N}$, and then passing to the $\mathrm{Airy}_1$ scaling limit at the GOE spectral edge. For the Ising SK model, the proof is based on a sphere-to-cube comparison principle showing that the quenched distributions of $N^{1/3} R_{1,2}$ under the spherical and Ising Gibbs measures asymptotically coincide. This paper is a companion to \cite{du2026fluctuations}, where we introduced a related comparison principle to identify the limiting fluctuations of the SK free energy. Most of the arguments in this paper were generated using GPT-5.6 Pro, with the aim of exploring further consequences of the ideas developed in that work.
Comments36 pages
