Rigidity and tolerance in point processes: Gaussian zeros and Ginibre eigenvalues

Abstract

Let Π be a translation invariant point process on the complex plane C and let D ⊂ C be a bounded open set. We ask what does the point configuration Π out obtained by taking the points of Π outside D tell us about the point configuration Π in of Π inside D? We show that for the Ginibre ensemble, Π out determines the number of points in Π in . For the translation-invariant zero process of a planar Gaussian Analytic Function, we show that Π out determines the number as well as the centre of mass of the points in Π in . Further, in both models we prove that the outside says "nothing more" about the inside, in the sense that the conditional distribution of the inside points, given the outside, is mutually absolutely continuous with respect to the Lebesgue measure on its supporting submanifold.

Rigidity of determinantal point processes with the Airy, the Bessel and the Gamma kernel

“…For the sine-process, rigidity is due to Ghosh [6]. For the Ginibre ensemble, rigidity has been established by Ghosh and Peres [7]; see also Osada and Shirai [16].…”

“…The following definition of rigidity of a point process is due to Ghosh [6] (cf. also Ghosh and Peres [7]). …”

“…We recall the sufficient condition for rigidity of a point process given by Ghosh [6], Ghosh and Peres [7]. [6], Ghosh and Peres [7]) Let P be a Borel probability measure on Conf(M).…”

“…[6], Ghosh and Peres [7]) Let P be a Borel probability measure on Conf(M). Assume that for any ε > 0 and any bounded subset B ⊂ M there exists a bounded measurable function f of bounded support such that f ≡ 1 on B and Var P S f < ε.…”

“…Proof For the reader's convenience, we recall the elegant short proof of Ghosh [6], Ghosh and Peres [7]. Let B (n) be an increasing sequence of nested bounded Borel sets exhausting M. Our assumptions and the Borel-Cantelli Lemma imply the existence of a sequence of bounded measurable function f (n) of bounded support, such that f (n) | B (n) ≡ 1 and that for P-almost every X ∈ Conf(M) we have…”