danbetea / tracy_widom

Tracy--Widom GUE and GOE distributions in Mathematica using the Bornemann Fredholm method

tracy_widom

This Mathematica project numerically computes the Tracy--Widom GUE (Gaussian Unitary Ensemble) and GOE (Gaussian Orthogonal Ensemble) cumulative distribution functions using the Fredholm determinant method of Bornemann.

A brief lay description of the Tracy--Widom distributions appears in this Quanta Magazine article. The GUE distribution first appeared in the work of Tracy and Widom, Level-Spacing Distributions and the Airy Kernel, arXiv:hep-th/9211141; and in the work of Forrester, The spectrum edge of random matrix ensembles, article available here (paywalled). The GOE distribution appeared in Tracy and Widom, On Orthogonal and Symplectic Matrix Ensembles, arXiv:solv-int/9509007.

The details of the numerical method can be found in the work of Bornemann, On the Numerical Evaluation of Fredholm Determinants, arXiv:0804.2543v2 math.NA.

The implementation agrees with the built-in Mathematica implementation (based on Painlevé transcendents as far as I know) to 6 digits save for the left tail (where the distribution function is exponentially close to 0). The method needs refinement to handle left tails properly.