Szegő Type Asymptotics for the Reproducing Kernel in Spaces of Full-Plane Weighted Polynomials
(2023) In Communications in Mathematical Physics 398(3). p.1291-1348- Abstract
Consider the subspace Wn of L2(C, dA) consisting of all weighted polynomials W(z)=P(z)·e-12nQ(z), where P(z) is a holomorphic polynomial of degree at most n- 1 , Q(z) = Q(z, z¯) is a fixed, real-valued function called the “external potential”, and dA=12πidz¯∧dz is normalized Lebesgue measure in the complex plane C. We study large n asymptotics for the reproducing kernel Kn(z, w) of Wn; this depends crucially on the position of the points z and w relative to the droplet S, i.e., the support of Frostman’s equilibrium measure in external potential Q. We mainly focus on the case when both z and w are in or near the component U of C^ \ S containing ∞, leaving aside such cases which are at this... (More)
Consider the subspace Wn of L2(C, dA) consisting of all weighted polynomials W(z)=P(z)·e-12nQ(z), where P(z) is a holomorphic polynomial of degree at most n- 1 , Q(z) = Q(z, z¯) is a fixed, real-valued function called the “external potential”, and dA=12πidz¯∧dz is normalized Lebesgue measure in the complex plane C. We study large n asymptotics for the reproducing kernel Kn(z, w) of Wn; this depends crucially on the position of the points z and w relative to the droplet S, i.e., the support of Frostman’s equilibrium measure in external potential Q. We mainly focus on the case when both z and w are in or near the component U of C^ \ S containing ∞, leaving aside such cases which are at this point well-understood. For the Ginibre kernel, corresponding to Q= | z| 2, we find an asymptotic formula after examination of classical work due to G. Szegő. Properly interpreted, the formula turns out to generalize to a large class of potentials Q(z); this is what we call “Szegő type asymptotics”. Our derivation in the general case uses the theory of approximate full-plane orthogonal polynomials instigated by Hedenmalm and Wennman, but with nontrivial additions, notably a technique involving “tail-kernel approximation” and summing by parts. In the off-diagonal case z≠ w when both z and w are on the boundary ∂U, we obtain that up to unimportant factors (cocycles) the correlations obey the asymptotic Kn(z,w)∼2πnΔQ(z)14ΔQ(w)14S(z,w)where S(z, w) is the Szegő kernel, i.e., the reproducing kernel for the Hardy space H02(U) of analytic functions on U vanishing at infinity, equipped with the norm of L2(∂U, | dz|). Among other things, this gives a rigorous description of the slow decay of correlations at the boundary, which was predicted by Forrester and Jancovici in 1996, in the context of elliptic Ginibre ensembles.
(Less)
- author
- Ameur, Yacin LU and Cronvall, Joakim LU
- organization
- publishing date
- 2023
- type
- Contribution to journal
- publication status
- published
- subject
- in
- Communications in Mathematical Physics
- volume
- 398
- issue
- 3
- pages
- 1291 - 1348
- publisher
- Springer
- external identifiers
-
- scopus:85142002652
- ISSN
- 0010-3616
- DOI
- 10.1007/s00220-022-04539-y
- language
- English
- LU publication?
- yes
- id
- 8dedd6ce-f720-4bd6-94c3-1777ddb13dc5
- date added to LUP
- 2023-01-20 12:08:15
- date last changed
- 2023-10-26 14:53:57
@article{8dedd6ce-f720-4bd6-94c3-1777ddb13dc5,
abstract = {{<p>Consider the subspace W<sub>n</sub> of L<sup>2</sup>(C, dA) consisting of all weighted polynomials W(z)=P(z)·e-12nQ(z), where P(z) is a holomorphic polynomial of degree at most n- 1 , Q(z) = Q(z, z¯) is a fixed, real-valued function called the “external potential”, and dA=12πidz¯∧dz is normalized Lebesgue measure in the complex plane C. We study large n asymptotics for the reproducing kernel K<sub>n</sub>(z, w) of W<sub>n</sub>; this depends crucially on the position of the points z and w relative to the droplet S, i.e., the support of Frostman’s equilibrium measure in external potential Q. We mainly focus on the case when both z and w are in or near the component U of C^ \ S containing ∞, leaving aside such cases which are at this point well-understood. For the Ginibre kernel, corresponding to Q= | z| <sup>2</sup>, we find an asymptotic formula after examination of classical work due to G. Szegő. Properly interpreted, the formula turns out to generalize to a large class of potentials Q(z); this is what we call “Szegő type asymptotics”. Our derivation in the general case uses the theory of approximate full-plane orthogonal polynomials instigated by Hedenmalm and Wennman, but with nontrivial additions, notably a technique involving “tail-kernel approximation” and summing by parts. In the off-diagonal case z≠ w when both z and w are on the boundary ∂U, we obtain that up to unimportant factors (cocycles) the correlations obey the asymptotic Kn(z,w)∼2πnΔQ(z)14ΔQ(w)14S(z,w)where S(z, w) is the Szegő kernel, i.e., the reproducing kernel for the Hardy space H02(U) of analytic functions on U vanishing at infinity, equipped with the norm of L<sup>2</sup>(∂U, | dz|). Among other things, this gives a rigorous description of the slow decay of correlations at the boundary, which was predicted by Forrester and Jancovici in 1996, in the context of elliptic Ginibre ensembles.</p>}},
author = {{Ameur, Yacin and Cronvall, Joakim}},
issn = {{0010-3616}},
language = {{eng}},
number = {{3}},
pages = {{1291--1348}},
publisher = {{Springer}},
series = {{Communications in Mathematical Physics}},
title = {{Szegő Type Asymptotics for the Reproducing Kernel in Spaces of Full-Plane Weighted Polynomials}},
url = {{http://dx.doi.org/10.1007/s00220-022-04539-y}},
doi = {{10.1007/s00220-022-04539-y}},
volume = {{398}},
year = {{2023}},
}