Spacing distributions in random matrix ensembles

The topic of spacing distributions in random matrix ensembles is almost as old as the introduction of random matrix theory into nuclear physics. Both events can be traced back to Wigner in the mid 1950’s [37, 38]. Thus Wigner introduced the model of a large real symmetric random matrix, in which the upper triangular elements are independently distributed with zero mean and constant variance, for purposes of reproducing the statistical properties of the highly excited energy levels of heavy nuclei. This was motivated by the gathering of experimental data on the spectrum of isotopes such as 238U at energy levels beyond neutron threshold. Wigner hypothesized that the statistical properties of the highly excited states of complex nuclei would be the same as those of the eigenvalues of large random real symmetric matrices. For the random matrix model to be of use at a quantitative level, it was necessary to deduce analytic forms of statistics of the eigenvalues which could be compared against statistics determined from experimental data. What are natural statistics for a sequence of energy levels, and can these statistics be computed for the random matrix model? Regarding the first question, let us think of the sequence as a point process on the line, and suppose for simplicity that the density of points is uniform and has been normalized to unity. For any point process in one dimension a fundamental quantity is the probability density function for the event that given there is a point at the origin, there is a point in the interval [s, s + ds], and further there are n points somewhere in between these points and thus in the interval (0, s). Let us denote the probability density function by p(n; s). In the language of energy levels, this is the spacing distribution between levels n apart. Another fundamental statistical quantity is the k-point distribution function ρ(k)(x1, . . . , xk). This can be defined recursively, starting with ρ(1)(x), by the requirement that