Abstract
Two optimal transport problems between determinantal point processes (DPP for short) are investigated. It is shown how to estimate the Kantorovitch–Rubinstein and Wasserstein-2 distances between distributions of DPP. These results are applied to evaluate the accuracy of a fast but approximate simulation algorithm of the Ginibre point process restricted to a circle. One can now simulate in a reasonable amount of time more than ten thousands points.
Highlights
Determinantal point processes (DPP) have been introduced in the seventies [24] to model fermionic particles with repulsion like electrons
We evaluate the impact of these approximations by bounding the distances between the original distribution of the DPP to be simulated and the distribution according to which the points are drawn
Let NE be the space of locally finite subsets of E, called the configuration space: NE = {ξ ⊂ E : | ∩ ξ | < ∞ for any compact set ⊂ E}, equipped with the topology of the vague convergence – the topology generated by the seminorms pf (ξ ) = f (x) x∈ξ for f : E → R continuous with compact support
Summary
Determinantal point processes (DPP) have been introduced in the seventies [24] to model fermionic particles with repulsion like electrons. They recently regained interest since they represent locations of eigenvalues of some random matrices. Following [18], the eigenvalues are not measurable functions of the realizations of the point process so it is difficult to devise how a modification of the eigenfunctions, respectively of the. We focus here on the total variation distance and on the quadratic Wasserstein distance The former counts the difference of the number of points in an optimal coupling between two distributions.
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