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The Polyanalytic Ginibre Ensembles
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).ORCID iD: 0000-0002-4971-7147
2013 (English)In: Journal of statistical physics, ISSN 0022-4715, E-ISSN 1572-9613, Vol. 153, no 1, 10-47 p.Article in journal (Refereed) Published
Abstract [en]

For integers n,q=1,2,3,aEuro broken vertical bar aEuro parts per thousand, let Pol (n,q) denote the -linear space of polynomials in z and , of degree a parts per thousand currency signn-1 in z and of degree a parts per thousand currency signq-1 in . We supply Pol (n,q) with the inner product structure of the resulting Hilbert space is denoted by Pol (m,n,q) . Here, it is assumed that m is a positive real. We let K (m,n,q) denote the reproducing kernel of Pol (m,n,q) , and study the associated determinantal process, in the limit as m,n ->+a while n=m+O(1); the number q, the degree of polyanalyticity, is kept fixed. We call these processes polyanalytic Ginibre ensembles, because they generalize the Ginibre ensemble-the eigenvalue process of random (normal) matrices with Gaussian weight. There is a physical interpretation in terms of a system of free fermions in a uniform magnetic field so that a fixed number of the first Landau levels have been filled. We consider local blow-ups of the polyanalytic Ginibre ensembles around points in the spectral droplet, which is here the closed unit disk . We obtain asymptotics for the blow-up process, using a blow-up to characteristic distance m (-1/2); the typical distance is the same both for interior and for boundary points of . This amounts to obtaining the asymptotical behavior of the generating kernel K (m,n,q) . Following (Ameur et al. in Commun. Pure Appl. Math. 63(12):1533-1584, 2010), the asymptotics of the K (m,n,q) are rather conveniently expressed in terms of the Berezin measure (and density) For interior points |z|< 1, we obtain that in the weak-star sense, where delta (z) denotes the unit point mass at z. Moreover, if we blow up to the scale of m (-1/2) around z, we get convergence to a measure which is Gaussian for q=1, but exhibits more complicated Fresnel zone behavior for q > 1. In contrast, for exterior points |z|> 1, we have instead that , where is the harmonic measure at z with respect to the exterior disk . For boundary points, |z|=1, the Berezin measure converges to the unit point mass at z, as with interior points, but the blow-up to the scale m (-1/2) exhibits quite different behavior at boundary points compared with interior points. We also obtain the asymptotic boundary behavior of the 1-point function at the coarser local scale q (1/2) m (-1/2).

Place, publisher, year, edition, pages
2013. Vol. 153, no 1, 10-47 p.
Keyword [en]
Bargmann-Fock space, Polyanalytic function, Determinantal point process
National Category
URN: urn:nbn:se:kth:diva-122140DOI: 10.1007/s10955-013-0813-xISI: 000323664300002ScopusID: 2-s2.0-84883561179OAI: diva2:620975
Swedish Research Council

QC 20150629

Available from: 2013-05-13 Created: 2013-05-13 Last updated: 2015-06-29Bibliographically approved
In thesis
1. Polyanalytic Bergman Kernels
Open this publication in new window or tab >>Polyanalytic Bergman Kernels
2013 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

The thesis consists of three articles concerning reproducing kernels ofweighted spaces of polyanalytic functions on the complex plane. In the first paper, we study spaces of polyanalytic polynomials equipped with a Gaussianweight. In the remaining two papers, more general weight functions are considered. More precisely, we provide two methods to compute asymptotic expansions for the kernels near the diagonal and then apply the techniques to get estimates for reproducing kernels of polyanalytic polynomial spaces equipped with rather general weight functions.

Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology, 2013. vii, 25 p.
Trita-MAT. MA, ISSN 1401-2278 ; 2013:02
Polyanalytic function, determinantal point process, Bergman kernel
National Category
Natural Sciences
urn:nbn:se:kth:diva-122073 (URN)
Public defence
2013-05-28, F3, Lindstedtsvägen 26, KTH, Stockholm, 13:00 (English)

QC 20130513

Available from: 2013-05-13 Created: 2013-05-08 Last updated: 2013-05-13Bibliographically approved

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Haimi, AnttiHedenmalm, Håkan
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