Singularity dominated strong fluctuations for some random matrix averages

Abstract

The circular and Jacobi ensembles of random matrices have their eigenvalue support on the unit circle of the complex plane and the interval $(0,1)$ of the real line respectively. The averaged value of the modulus of the corresponding characteristic polynomial raised to the power $2\mu$ diverges, for $2\mu \le -1$, at points approaching the eigenvalue support. Using the theory of generalized hypergeometric functions based on Jack polynomials, the functional form of the leading asymptotic behaviour is established rigorously. In the circular ensemble case this confirms a conjecture of Berry and Keating.

11 pages, to appear Commun. Math. Phys

References in corpus (1)

• Negative moments of characteristic polynomials of random GOE matrices and singularity-dominated strong fluctuations

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