Patterns in eigenvalues: the 70th Josiah Willard Gibbs lecture. (English) Zbl 1161.15302
Summary: Typical large unitary matrices show remarkable patterns in their eigenvalue distribution. These same patterns appear in telephone encryption, the zeros of Riemann’s zeta function, a variety of physics problems, and in the study of Toeplitz operators. This paper surveys these applications and what is currently known about the patterns.
MSC:
| 15A18 | Eigenvalues, singular values, and eigenvectors |
| 60B15 | Probability measures on groups or semigroups, Fourier transforms, factorization |
| 82B31 | Stochastic methods applied to problems in equilibrium statistical mechanics |
| 82B44 | Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics |
| 15B52 | Random matrices (algebraic aspects) |
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