Abstract
We investigate the nonperturbative physics of the zero-dimensional random Hermitian matrix model, using semiclassical analysis as well as orthogonal polynomials. Finite-N tunneling leads to a unique equilibration of the Dyson gas of eigenvalues and dissolves a fictitious family of N=∞ saddle points. We present a mean-field potential for the limiting multiple-arc eigenvalue distribution. The sequence of the orthogonal-polynomial recursion coefficients Rk is characterized by the critical points of the matrix potential. Its large-N limit can show regions of smooth, quasi-periodic and seemingly chaotic behavior. The tunneling competition between nondegenerate potential wells is the origin of the unpredictability.
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