On the Numerical Evaluation of Distributions in Random Matrix Theory: A Review

#### F. Bornemann

2010, v.16, №4, 803-866

ABSTRACT

In this paper we review and compare the numerical evaluation of those probability distributions in random matrix theory that are analytically represented in terms of Painleve transcendents or Fredholm determinants. Concrete examples for the Gaussian and Laguerre (Wishart) $\beta$-ensembles and their various scaling limits are discussed. We argue that the numerical approximation of Fredholm determinants is the conceptually more simple and efficient of the two approaches, easily generalized to the computation of joint probabilities and correlations. Having the means for extensive numerical explorations at hand, we discovered new and surprising determinantal formulae for the k-th largest (or smallest) level in the edge scaling limits of the Orthogonal and Symplectic Ensembles; formulae that in turn led to improved numerical evaluations. The paper comes with a toolbox of Matlab functions that facilitates further mathematical experiments by the reader.

Keywords: random matrix theory,numerical approximation,Painleve transcendents,Fredholm determinants