Baron Joseph W, Jewell Thomas Jun, Ryder Christopher, Galla Tobias
Instituto de Física Interdisciplinar y Sistemas Complejos IFISC (CSIC-UIB), 07122 Palma de Mallorca, Spain.
Department of Physics and Astronomy, School of Natural Sciences, The University of Manchester, Manchester M13 9PL, United Kingdom.
Phys Rev Lett. 2022 Mar 25;128(12):120601. doi: 10.1103/PhysRevLett.128.120601.
Random matrix theory allows one to deduce the eigenvalue spectrum of a large matrix given only statistical information about its elements. Such results provide insight into what factors contribute to the stability of complex dynamical systems. In this Letter, we study the eigenvalue spectrum of an ensemble of random matrices with correlations between any pair of elements. To this end, we introduce an analytical method that maps the resolvent of the random matrix onto the response functions of a linear dynamical system. The response functions are then evaluated using a path integral formalism, enabling us to make deductions about the eigenvalue spectrum. Our central result is a simple, closed-form expression for the leading eigenvalue of a large random matrix with generalized correlations. This formula demonstrates that correlations between matrix elements that are not diagonally opposite, which are often neglected, can have a significant impact on stability.
随机矩阵理论允许人们仅根据关于矩阵元素的统计信息来推导大型矩阵的特征值谱。此类结果有助于深入了解哪些因素对复杂动力系统的稳定性有贡献。在本信函中，我们研究了元素对之间存在相关性的随机矩阵系综的特征值谱。为此，我们引入一种解析方法，该方法将随机矩阵的预解式映射到线性动力系统的响应函数上。然后使用路径积分形式来评估响应函数，从而使我们能够对特征值谱进行推导。我们的核心结果是一个关于具有广义相关性的大型随机矩阵的主导特征值的简单闭式表达式。该公式表明，通常被忽略的非对角相对矩阵元素之间的相关性会对稳定性产生重大影响。
