Sampling from Sine Kernel and Airy Kernel

Just in case someone is still interested. The algorithm of Hough, Krishnapur, Peres and Virag was implemented here

http://arxiv.org/abs/1404.0071

in the case of eigenvalues of random matrices. The methodology uses Chebyshev approximations to do inverse sampling of the marginal distributions. The code is available here in the Julia RMT package:

https://github.com/jiahao/RandomMatrices.jl

For a general algorithm for simulating points from a determinantal process, see Algorithm 18 in the paper "Determinantal Processes and Independence" by Hough, Krishnapur, Peres and Virag:

arXiv link

This algorithm was actually implemented by some physicists at Princeton (I believe) but I am not sure if their code is publicly available.

For the sine kernel, depending on how many points you want to sample, Matlab is pretty good at computing eigenvalues of a large GUE matrix in a decently short amount of time. That would require much less work than implementing the algorithm above.