利用源自确定性与随机微分方程的互相关随机矩阵识别模式
Identifying Patterns Using Cross-Correlation Random Matrices Derived from Deterministic and Stochastic Differential Equations
AI总结：
本研究提出利用由确定性或随机微分方程推导的映射构建的互相关随机矩阵的谱特性，来指示系统的临界或混沌行为，无需蒙特卡罗模拟等技术。
摘要：
互相关随机矩阵已成为自旋系统相变的一种颇具前景的指示指标。其核心概念是磁化强度的演化蕴含热力学信息[R. da Silva, Int. J. Mod. Phys. C, 2350061 (2023)]，该信息直接反映在这些矩阵的特征值中。当在平均场区域分析这些演化过程时，一个重要问题随之产生：朗之万方程（Langevin equation）转化为映射后，能否发挥相同作用？已有部分研究表明该方法或许也能捕捉特定系统的混沌行为。本研究提出，由确定性或随机微分方程推导得到的映射所构建的随机矩阵的谱特性，可指示此类系统的临界行为或混沌行为。对于混沌系统，我们仅需迭代哈密顿方程（Hamiltonian equations）的演化过程；对于自旋系统，由平均场方程得到的朗之万映射（Langevin maps）便已足够，由此无需开展蒙特卡罗（MC）模拟或采用其他技术。
英文摘要：
Cross-Correlation random matrices have emerged as a promising indicator of phase transitions in spin systems. The core concept is that the evolution of magnetization encapsulates thermodynamic information [R. da Silva, Int. J. Mod. Phys. C, 2350061 (2023)], which is directly reflected in the eigenvalues of these matrices. When these evolutions are analyzed in the mean-field regime, an important question arises: Can the Langevin equation, when translated into maps, perform the same function? Some studies suggest that this method may also capture the chaotic behavior of certain systems. In this work, we propose that the spectral properties of random matrices constructed from maps derived from deterministic or stochastic differential equations can indicate the critical or chaotic behavior of such systems. For chaotic systems, we need only the evolution of iterated Hamiltonian equations, and for spin systems, the Langevin maps obtained from mean-field equations suffice, thus avoiding the need for Monte Carlo (MC) simulations or other techniques.
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