具有广义相关性的随机矩阵的特征值：路径积分方法
Eigenvalues of random matrices with generalised correlations: a path integral approach
AI总结：
本文提出一种基于路径积分的解析方法，将随机矩阵预解式映射为线性系统响应函数，推导出具有广义相关性的大型随机矩阵主特征值的闭式表达式，揭示了非对角元素关联对系统稳定性的显著影响。
摘要：
随机矩阵理论允许人们在仅给定元素统计信息的情况下推导大矩阵的特征值谱。这些结果为哪些因素影响复杂动力系统的稳定性提供了深刻见解。在这封信中，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，我们引入了一种解析方法，将随机矩阵的预解式映射到线性动力系统的响应函数上。随后使用路径积分形式计算这些响应函数，使我们能够对特征值谱进行推断。我们的核心结果是针对具有广义相关性的大型随机矩阵的主特征值的一个简单的闭式表达式。该公式表明，矩阵元素之间非对角的关联性往往被忽略，但它们能对稳定性产生显著影响。
英文摘要：
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 generalised 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.
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