Random Matrices and Free Probability

Random matrix theory studies the statistical properties of eigenvalues and eigenvectors of large matrices with random entries. When the entries are independent and the matrix is scaled by $\frac{1}{\sqrt{N}}$, the empirical spectral distribution converges to the Wigner semicircle law. This talk surveys recent progress in proving universality — the remarkable phenomenon that local eigenvalue statistics depend only on the symmetry class of the matrix and not on the distribution of its entries. We explain how these results connect to Voiculescu's free probability theory, where classical independence is replaced by free independence, and how free convolution provides a powerful framework for understanding the limiting behavior of sums and products of large random matrices.