Macroscopic analysis of determinantal random balls

Abstract

We consider a collection of Euclidean random balls in R d generated by a determinantal point process inducing interaction into the balls. We study this model at a macroscopic level obtained by a zooming-out and three different regimes -Gaussian, Poissonian and stable-are exhibited as in the Poissonian model without interaction. This shows that the macroscopic behaviour erases the interactions induced by the determinantal point process. arXiv:1504.04513v2 [math.PR] 1 Jun 20171 Determinantal random balls modely − x ≤ r whose centers x ∈ R d and radii r ∈ R + are generated by a marked stationary determinantal point process (Dpp) Φ on R d × R + . In this section, we describe thoroughly the model and we refer to the Appendix A for more details on Dpp, in particular see the definition in Def. A.2. First, consider a stationary Dpp φ with a kernel K with respect to the Lebesgue measure Leb satisfying K(x, y) = K(x − y) (for simplicity, we use the same letter K for two different functions), moreover we assume that the map K given for all x ∈ R d and any f ∈ L 2 (R d , dx)satisfies the following hypothesis Hypothesis 1 The map K in (3) is a bounded symmetric integral operator K from L 2 (R d , dx) into L 2 (R d , dx) with spectrum included in [0, 1[. Moreover, K is locally trace-class, i.e. for all compact Λ ⊂ E, the restriction K Λ of K on L 2 (Λ, λ) is of trace-class.

Rescaled weighted determinantal random balls

“…Now, we can state the main result of this note. The proof consists in a combination of the arguments of [3] and [6]. It is given in Section 3 where for some required technical points, it is referred to [3] and [4].…”

“…The asymptotics condition in (5) is of constant use in the following. Proof: The proof follows the same lines as that of Proposition 1.1 in [3], replacing K(0) by K(x, x) and controlling it thanks to Hypothesis (2).…”

“…The next step is to consider repulsion between the balls. In [3], Breton, Clarenne and Gobard give results on determinantal random balls model, but no weight are considered in their model. In this note, we consider weighted random balls generated by a non-stationary determinantal point process.…”

“…To that purpose, we give an extension of the Laplace transform of determinantal processes allowing to compute Laplace transform with not necessarily compactly supported function, but with instead a condition of integrability. The main contributions of this note thus are a simplication of the proof of [3] and the introduction of weights in the non-stationary determinantal random balls model. This paper is organized as follows.…”

Macroscopic analysis of shot-noise Cox random balls

“…We perform a scaling in this model by first shrinking the radii; to compensate this effect, we rescale the shot-noise Cox process Z that generated the centers of the balls. In contrast with the Poissonian (see [1]) or the determinantal case (see [3]), where there is just one level of randomness, the global location of the balls, here we have two levels of randomness with the collection of cluster and within each cluster. As a consequence, this additional level of randomness makes it possible to study many more different asymptotics behaviours in this model.…”