MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4).

## Sampling from Sine Kernel and Airy Kernel

A determinantal process on the line is a random collection of points on $\mathbb{R}$ such that the probability of $x_1, \dots, x_n$ lying on the random set is $\det (K(x_i, x_j))_{(i,j)}$. Examples of determinantal processes include the eigenvalues of random Hermitean matrices with Gaussian entries and non-intersecting random walks. I'm interested in sampling random points on the line according to the sine kernel $k(x,y) = \frac{\sin(x-y)}{\pi(x-y)}$ or the Airy Kernel (see p 6 of the slides) which are related to the Gaussian Unitary Ensemble.

I thought there are algorithms for sampling from general discrete and continuous determinantal processes. Maybe it's be better to sample a processes directly using Coupling from the Past and other procedures.

Mainly,

• how does one sample points on the real line with respect to the sine kernel?
• is there a general way of sampling determinantal processes based or arbitrary kernel?

It is known these types of processes demonstrate repulsion (compared to the Poisson process) and I would like to demonstrate this in the classroom.

-

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: