Random matrices and random Schroedinger operators

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Random matrices and random Schroеdinger operators. In the 1950s, Wigner initiated a program to understand physical phenomena through the eigenvalues of random matrices. Dyson called this the “new statistical mechanics”. While it has been a remarkably successful approach in physics, almost nothing has been proven mathematically.

I will show how to construct a “continuum limit” of the Gaussian random matrix ensembles. A similar approach can be applied to a di?erent physical model, random Schroеdinger operators. In certain cases, we can verify the physics predictions: the eigenvalue distributions agree.