Gaussian point processes and two-by-two random matrix theory

Abstract

The statistics of the multidimensional Gaussian point process are discussed in connection with the spacing statistics of eigenvalues of 2x2 random matrices. We consider the three-dimensional Gaussian point process when two of the coordinates of a point are randomly chosen from a Gaussian distribution having a mean of zero and a variance of sigma;{2}=1 but the third coordinate is chosen from a Gaussian distribution having a variance in the range of 0< or =sigma_{3};{2}< or =1 . The probability density function associated with a random point being at a distance r from the origin is shown to be closely related to the nearest-neighbor spacing distribution of eigenvalues coming from an ensemble of 2x2 matrices defined by the French-Kota-Pandey-Mehta two-matrix model of random matrix theory. An elementary explanation of this result is given.