纯球形自旋玻璃的低温朗温动力学阈值能

IF 3.1 1区数学 Q1 MATHEMATICS

Communications on Pure and Applied Mathematics Pub Date : 2024-04-19 DOI:10.1002/cpa.22197

Mark Sellke

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引用次数: 0

摘要

我们研究了球形-自旋模型的朗格文动力学，重点是库里安多洛-库尔坎方程所描述的短时间机制。我们证实了 Cugliandolo 和 Kurchan 的预言，并证明所获得的渐近能量恰好处于低温极限。上界使用了 Lipschitz 优化算法的硬度结果，适用于所有温度。对于下限，我们证明了动力学达到并保持在任何近似局部最大值的最低能量之上。事实上，后一种行为适用于任何服从自然平滑估计的哈密顿，即使是在无序初始化和指数时间尺度上也是如此。

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The threshold energy of low temperature Langevin dynamics for pure spherical spin glasses

We study the Langevin dynamics for spherical $p$ -spin models, focusing on the short time regime described by the Cugliandolo–Kurchan equations. Confirming a prediction of Cugliandolo and Kurchan, we show the asymptotic energy achieved is exactly $E_{\infty} (p) = 2 \sqrt{\frac{p - 1}{p}}$ in the low temperature limit. The upper bound uses hardness results for Lipschitz optimization algorithms and applies for all temperatures. For the lower bound, we prove the dynamics reaches and stays above the lowest energy of any approximate local maximum. In fact the latter behavior holds for any Hamiltonian obeying natural smoothness estimates, even with disorder-dependent initialization and on exponential time-scales.

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来源期刊

Communications on Pure and Applied Mathematics 数学-数学

CiteScore

6.70

自引率

3.30%

发文量

审稿时长

>12 weeks

期刊介绍： Communications on Pure and Applied Mathematics (ISSN 0010-3640) is published monthly, one volume per year, by John Wiley & Sons, Inc. © 2019. The journal primarily publishes papers originating at or solicited by the Courant Institute of Mathematical Sciences. It features recent developments in applied mathematics, mathematical physics, and mathematical analysis. The topics include partial differential equations, computer science, and applied mathematics. CPAM is devoted to mathematical contributions to the sciences; both theoretical and applied papers, of original or expository type, are included.