Gap Formation Probability of the $\alpha -$ Ensemble

Abstract

In this paper we employ the continuum approximation of Dyson to determine the asymptotic gap formation probability in the spectrum of $N\times N$ Hermitean random matrices. The associated orthogonal polynomials has weight function, $w(x)=e^{-u(x)},$ where $u(x)=x^{\alpha },\alpha >0,0<x<\infty .$ It is shown that the probability that the scaled interval $(0,s)$ contains no eigenvalues is universal for $\alpha >1/2$ and depends on $\alpha$ for $0<\alpha <1/2.$

29, latex

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