Pastur's theorem

Pastur's theorem describes how the behavior of large, complex systems like disordered materials or large random matrices can be understood through their spectral (eigenvalue) distribution. It states that as the size of such systems grows infinitely large, the distribution of their eigenvalues becomes deterministic and follows a specific, predictable pattern known as the Pastur distribution or semi-circular law in some cases. This theorem helps scientists analyze phenomena in physics, signal processing, and mathematics by simplifying the complexity, showing that despite randomness at the microscopic level, the macroscopic behavior exhibits regularity and can be precisely characterized.