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Polyanalytic Bergman Kernels
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
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.
Series
Trita-MAT. MA, ISSN 1401-2278 ; 2013:02
Keyword [en]
Polyanalytic function, determinantal point process, Bergman kernel
National Category
Natural Sciences
Identifiers
URN: urn:nbn:se:kth:diva-122073OAI: oai:DiVA.org:kth-122073DiVA: diva2:620524
Public defence
2013-05-28, F3, Lindstedtsvägen 26, KTH, Stockholm, 13:00 (English)
Opponent
Supervisors
Note

QC 20130513

Available from: 2013-05-13 Created: 2013-05-08 Last updated: 2013-05-13Bibliographically approved
List of papers
1. The Polyanalytic Ginibre Ensembles
Open this publication in new window or tab >>The Polyanalytic Ginibre Ensembles
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).

Keyword
Bargmann-Fock space, Polyanalytic function, Determinantal point process
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-122140 (URN)10.1007/s10955-013-0813-x (DOI)000323664300002 ()2-s2.0-84883561179 (Scopus ID)
Funder
Swedish Research Council
Note

QC 20150629

Available from: 2013-05-13 Created: 2013-05-13 Last updated: 2017-12-06Bibliographically approved
2. Asymptotic expansion of polyanalytic Bergman kernels
Open this publication in new window or tab >>Asymptotic expansion of polyanalytic Bergman kernels
2014 (English)In: Journal of Functional Analysis, ISSN 0022-1236, E-ISSN 1096-0783, Vol. 267, no 12, 4667-4731 p.Article in journal (Refereed) Published
Abstract [en]

We consider the q-analytic functions on a given planar domain Omega, square integrable with respect to a weight. This gives us a q-analytic Bergman kernel, which we use to extend the Bergman metric to this context. We recall that f is q-analytic if (partial derivative) over bar (q) f = 0 for the given positive integer q. Polyanalytic Bergman spaces and kernels appear naturally in time-frequency analysis of Gabor systems of Hermite functions as well as in the mathematical physics of the analysis of Landau levels.

We obtain asymptotic formulae in the bulk for the q-analytic Bergman kernel in the setting of the power weights e(-2mQ), as the positive real parameter m tends to infinity. This is only known previously for q = 1, by the work of Tian, Yau, Zelditch, and Catlin. Our analysis, however, is inspired by the more recent approach of Berman, Berndtsson, and Sjostrand, which is based on ideas from microlocal analysis.

We remark here that since a q-analytic function may be identified with a vector-valued holomorphic function, the Bergman space of q-analytic functions may be understood as a vector-valued holomorphic Bergman space supplied with a certain singular local metric on the vectors. Finally, we apply the obtained asymptotics for q = 2 to the bianalytic Bergman metrics, and after suitable blow-up, the result is independent of Q for a wide class of potentials Q. We interpret this as an instance of geometric universality.

Keyword
Polyanalytic functions, Bergman kernel, Asymptotic expansion, Bulk universality
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-122142 (URN)10.1016/j.jfa.2014.09.002 (DOI)000346226500003 ()2-s2.0-84921969223 (Scopus ID)
Funder
Göran Gustafsson Foundation for Research in Natural Sciences and MedicineSwedish Research Council, 2012-3122
Note

QC 20150220. Updated from submitted to published.

Available from: 2013-05-13 Created: 2013-05-13 Last updated: 2017-12-06Bibliographically approved
3. Bulk asymptotics for polyanalytic correlation kernels
Open this publication in new window or tab >>Bulk asymptotics for polyanalytic correlation kernels
2014 (English)In: Journal of Functional Analysis, ISSN 0022-1236, E-ISSN 1096-0783, Vol. 266, no 5, 3083-3133 p.Article in journal (Refereed) Published
Abstract [en]

For a weight function Q : C -> R and a positive scaling parameter in, we study reproducing kernels K-q,K-mQ,K-n of the polynomial spaces A(q,mQ,n)(2) :=span(C) {(z) over bar (-r) z(j) vertical bar 0 <= r <= q-1, 0 <= j <= n-1} equipped with the inner product from the space L-2 (e(-mQ(z)) dA(z)). Here dA denotes a suitably normalized area measure on C. For a point z(0) belonging to the interior of certain compact set S and satisfying Delta Q (z(0)) > 0, we define the resealed coordinates z = z(0) + xi/root m Delta Q(z(0)), w=z(0) + lambda/root m Delta Q(z(0)). The following universality result is proved in the case q = 2: 1/m Delta Q(z(0))vertical bar K-q,K-mQ,K-n(z,w)vertical bar e(-1/2mQ(z)-1/2mQ(w)) -> vertical bar L-q-1(1) (vertical bar xi - lambda vertical bar(2))vertical bar e(-1/2 vertical bar xi-lambda vertical bar 2) as m,n -> infinity while n >= m - M for any fixed M > 0, uniformly for (xi,lambda) in compact subsets of C-2. The notation L-q-1(1) stands for the associated Laguerre polynomial with parameter 1 aid degree q - 1. This generalizes a result of Ameur, Hedenmalm and Makarov concerning analytic polynomials to bianalytic polynomials. We also discuss how to generalize the result to q > 2. Our methods include a simplification of a Bergman kernel expansion algorithm of Berman; Bemdtsson and Sjostrand in the one compex variable setting, and extension to the context of polyanalytic functions. We also study off-diagonal behaviour of the kernels K-q,K-mQ,K-n.

Keyword
Polyanalytic function, Determinantal point process, Landau level, Bergman kernel
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-122148 (URN)10.1016/j.jfa.2013.11.021 (DOI)000330928200015 ()2-s2.0-84893684275 (Scopus ID)
Note

QC 20140306. Updated from submitted to published.

Available from: 2013-05-13 Created: 2013-05-13 Last updated: 2017-12-06Bibliographically approved

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