A short introduction to operator limits of random matrices

These are notes to a four-lecture minicourse given at the 2017 PCMI Summer Session on Random Matrices. We give a quick introduction to the theory of large random matrices by taking limits that preserve their operator structure, rather than just their eigenvalues. The operator structure takes the role of exact formulas, and allows for results in the context of general β-ensembles. Along the way, we cover a non-computational proof of the Wiegner semicircle law, a quick proofs for the Füredi-Komlós result on the top eigenvalue, as well as the BBP phase transition. 1. The Gaussian Ensembles 1.1. The Gaussian Orthogonal and Unitary Ensembles. One of the earliest appearances of random matrices in mathematics was due to Eugene Wigner in the 1950’s. Let G be an n×nmatrix with independent standard normal entries. Then Mn = G+Gt √ 2 . This distribution on symmetric matrices is called the Gaussian Orthogonal Ensemble, because it is invariant under orthogonal conjugation. For any orthogonal matrix OMnO has the same distribution as Mn. To check this, note that OG has the same distribution as G be the rotation invariance of the Gaussian column vectors, and the same is true for OGO−1 by the rotation invariance of the row vectors. To finish note that orthogonal conjugation commutes with symmetrization. If we instead start with a matrix with independent standard complex Gaussian entries, we get the Gaussian Unitary ensemble. To see how the eigenvalues behave, we recall the following classical theorem. Theorem 1.1.1. Suppose Mn has GOE or GUE distribution then Mn has eigenvalue density (1.1.2) f(λ1, ..., λn) = 1 Zn n ∏