The Probability of an Eigenvalue Number Fluctuation in an Interval of a Random Matrix Spectrum

Abstract

We calculate the probability to find exactly $n$ eigenvalues in a spectral interval of a large random $N\times N$ matrix when this interval contains $s\ll N$ eigenvalues on average. The calculations exploit an analogy to the problem of finding a two-dimensional charge distribution on the interface of a semiconductor heterostructure under the influence of a split gate.

4 pages, postscript

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