A Universal Matrix Ensemble that Unifies Eigenspectrum Laws via Neural Network Models

Abstract: Random matrix theory, which characterizes the spectrum distribution of infinitely large matrices, plays a central role in theories across diverse fields, including high-dimensional data analysis, ecology, neuroscience, and machine learning. Among its celebrated achievements, the Marchenko--Pastur law and the elliptic law have served as key results for numerous applications. However, the relationship between these two laws remains elusive, and the existence of a universal framework unifying them is unclear. Inspired by a neural network model, we establish a universal matrix ensemble that unifies these laws as special cases. Through an analysis based on the saddle-node equation, we derive an explicit expression for the spectrum distribution of the ensemble. As a direct application, we reveal how the universal law clarifies the stability of a class of associative memory neural networks. By uncovering a fundamental law of random matrix theory, our results deepen the understanding of high-dimensional systems and advance the integration of theories across multiple disciplines.