👉 The Anderson-Miller (AM) conjecture, proposed in 1999, is a fundamental concept in the study of random matrix theory and its applications to physics and mathematics. It posits that for a large ensemble of independent, identically distributed (i.i.d.) random matrices, the eigenvalue distribution of certain submatrices converges to a universal limiting distribution known as the circular law, provided the matrix entries have zero mean and finite variance. Specifically, for a large random matrix with i.i.d. entries on the unit disk, the eigenvalues tend to fill the unit disk uniformly in the complex plane as the matrix size grows. This conjecture is significant because it provides a precise description of eigenvalue statistics in random matrices, bridging the gap between statistical mechanics and number theory, and has profound implications for understanding complex systems in physics, quantum chaos, and data science.
