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Joint probability densities of level spacing ratios in random matrices. (English) Zbl 1275.81044

Summary: We calculate analytically, for finite-size matrices, joint probability densities of ratios of level spacings in ensembles of random matrices characterized by their associated confining potential. We focus on the ratios of two spacings between three consecutive real eigenvalues, as well as certain generalizations such as the overlapping ratios. The resulting formulas are further analyzed in detail in two specific cases: the \(\beta\)-Hermite and the \(\beta\)-Laguerre cases, for which we offer explicit calculations for small \(N\). The analytical results are in excellent agreement with numerical simulations of usual random matrix ensembles, and with the level statistics of a quantum many-body lattice model and zeros of the Riemann zeta function.

MSC:

81Q50 Quantum chaos
81V70 Many-body theory; quantum Hall effect
15B52 Random matrices (algebraic aspects)
15A18 Eigenvalues, singular values, and eigenvectors
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
11M38 Zeta and \(L\)-functions in characteristic \(p\)
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