发表机构
Yale University（耶鲁大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究\(\mathbb{R}^n\)上通过对称矩阵\(X\)相互作用的扩散过程，推导样本路径经验律的动力学平均场极限，分析过阻尼朗之万扩散的平均场极限，得出自由能在特定条件下收敛到 replica - 对称极限，确定性模型满足条件时也成立。
AI 中文摘要
我们研究了一类在\(\mathbb{R}^n\)上通过对称矩阵\(X\in\mathbb{R}^{n\times n}\)相互作用的扩散过程。当\(X\)的特征向量在正交群上是哈尔均匀分布时，我们推导了样本路径经验律的动力学平均场极限，扩展了\(X\sim\mathrm{GOE}\)时的经典 Sompolinsky - Zippelius 特征。该极限采用具有相关高斯噪声和记忆的广义朗之万方程形式，其相关和响应核通过涉及\(X\)特征值分布自由累积量的卷积方程与原始动力学的相关和响应核相关。对于与\(\mu(\boldsymbol{\theta})\propto \exp\!\big(\frac12\boldsymbol{\theta}^{\top}X\boldsymbol{\theta}\big)\prod_{i=1}^n\nu(\mathrm{d}\theta_i)\)相关的过阻尼朗之万扩散，我们在快速混合假设下分析了平均场极限。相关和响应核允许满足涨落耗散关系的时间平移不变近似。广义朗之万方程允许与辅助多元 OU 过程耦合的马尔可夫近似，并收敛到\(\mu\)下经验坐标律的 replica - 对称预测。这种辅助相关结构通过提升路径历史过程的马尔可夫半群的无穷小生成元来表征。因此，在明确的高温条件下，自由能收敛到 replica - 对称极限，对于伊辛模型，该条件为\(\|X\|_{\mathrm{op}}<1/2\)。根据最近的动力学普遍性结果，当\(X\)满足一组确定性离域条件时，对于无随机无序的确定性模型，相同的自由能特征也成立。
英文摘要
We study a class of diffusion processes on $\mathbb{R}^n$ interacting through a symmetric matrix $X\in\mathbb{R}^{n\times n}$. When eigenvectors of $X$ are Haar-uniform on the orthogonal group, we derive a dynamical mean-field limit for the empirical law of sample paths, extending the classical Sompolinsky--Zippelius characterization for $X\sim\mathrm{GOE}$. The limit takes the form of a generalized Langevin equation with correlated Gaussian noise and memory, whose correlation and response kernels relate to those of the original dynamics through convolution equations involving the free cumulants of the eigenvalue distribution of $X$. For the overdamped Langevin diffusion associated with $μ(\boldsymbolθ)\propto \exp\!\big(\frac12\boldsymbolθ^{\top}X\boldsymbolθ\big)\prod_{i=1}^nν(\mathrm{d}θ_i)$, we analyze the mean-field limit under a rapid-mixing assumption. The correlation and response kernels admit time-translation-invariant approximants satisfying a fluctuation-dissipation relation. The generalized Langevin equation admits a Markovian approximation coupled to an auxiliary multivariate OU process and converges to a replica-symmetric prediction for the empirical coordinate law under $μ$. This auxiliary correlation structure is characterized through the infinitesimal generator of a Markov semigroup for the lifted path-history process. Consequently, the free energy converges to a replica-symmetric limit under an explicit high-temperature condition, which for an Ising model is $\|X\|_{\mathrm{op}}<1/2$. By recent dynamical universality results, the same free-energy characterization holds for deterministic models without random disorder when $X$ satisfies a set of deterministic delocalization conditions.
