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Bulk asymptotics for polyanalytic correlation kernels
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
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.

Place, publisher, year, edition, pages
2014. Vol. 266, no 5, 3083-3133 p.
Keyword [en]
Polyanalytic function, Determinantal point process, Landau level, Bergman kernel
National Category
URN: urn:nbn:se:kth:diva-122148DOI: 10.1016/j.jfa.2013.11.021ISI: 000330928200015ScopusID: 2-s2.0-84893684275OAI: diva2:621037

QC 20140306. Updated from submitted to published.

Available from: 2013-05-13 Created: 2013-05-13 Last updated: 2014-03-06Bibliographically 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, Antti
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