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

TL;DR

This paper calculates the probability of finding exactly n eigenvalues within a spectral interval of a large random matrix, using an analogy to charge distribution problems in semiconductor heterostructures.

Contribution

It introduces a novel approach to compute eigenvalue fluctuation probabilities in large random matrices by leveraging an analogy with charge distribution models.

Findings

Derived explicit probability formulas for eigenvalue counts in spectral intervals.

Established a connection between random matrix eigenvalues and semiconductor charge distributions.

Provided insights into eigenvalue fluctuation behavior in large matrices.

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.