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Sampling from Determinantal ProcessesSine 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 is are related to the Gaussian Unitary Ensemble. However, I'm sure

I thought there are algorithms for sampling from general discrete and continuous determinantal processes. On the other hand, it might Maybe it's be better to sample a processes like the non-intersecting random walks directly and the dimer model is sampled using Coupling from the Past and other procedures.

Mainly,

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

It is known these types of processes demonstrate repulsion (compared to the Poisson process) and I would like to demonstrate this empiricallyin the classroom.
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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 $k(x,y) = \frac{\sin(x-y)}{\pi(x-y)}$ which is related to the Gaussian Unitary Ensemble. However, I'm sure there are algorithms for sampling from general discrete and continuous determinantal processes. On the other hand, it might be better to sample a processes like the non-intersecting random walks directly and the dimer model is sampled using Coupling from the Past.

Mainly, how does one sample points on the real line with respect to the sine kernel? Secondly, is there a general way of sampling determinantal processes based on their kernel? It is known these types of processes demonstrate repulsion (compared to the Poisson process) and I would like to demonstrate this empirically.
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Sampling from Determinantal Processes

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 $k(x,y) = \frac{\sin(x-y)}{\pi(x-y)}$ which is related to the Gaussian Unitary Ensemble. However, I'm sure there are algorithms for sampling from general discrete and continuous determinantal processes. On the other hand, it might be better to sample a processes like the non-intersecting random walks directly and the dimer model is sampled using Coupling from the Past.

Mainly, how does one sample points on the real line with respect to the sine kernel? Secondly, is there a general way of sampling determinantal processes based on their kernel?