| 1. |
- Ameur, Yacin, et al.
(författare)
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Berezin Transform in Polynomial Bergman Spaces
- 2010
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Ingår i: Communications on Pure and Applied Mathematics. - : Wiley. - 0010-3640 .- 1097-0312. ; 63:12, s. 1533-1584
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Tidskriftsartikel (refereegranskat)abstract
- Fix a smooth weight function Q in the plane, subject to a growth condition from below Let K-m,K-n denote the reproducing kernel for the Hilbert space of analytic polynomials of degree at most n - 1 of finite L-2-norm with respect to the measure e-(mQ) dA Here dA is normalized area measure, and m is a positive real scaling parameter The (polynomial) Berezin measure dB(m,n)(< z0 >) (z) = K-m,K-n(z(0).z(0))(-1) vertical bar K-m,K-n(z.z(0))vertical bar(2)e(-mQ(z)) dA(z) for the point z(0) is a probability measure that defines the (polynomial) Berezin transform B-m,B-n f(z(0)) = integral(C) f dB(m,n)(< z0 >) for continuous f is an element of L-infinity (C). We analyze the semiclassical limit of the Berezin measure (and transform) as m -> +infinity while n = m tau + o(1), where tau is fixed, positive, and real We find that the Berezin measure for z(0) converges weak-star to the unit point mass at the point z(0) provided that Delta Q(z(0)) > 0 and that z(0) is contained in the interior of a compact set f(tau). defined as the coincidence set for an obstacle problem. As a refinement, we show that the appropriate local blowup of the Berezin measure converges to the standardized Gaussian measure in the plane For points z(0) is an element of C\f(tau), the Berezin measure cannot converge to the point mass at z(0) In the model case Q(z) = vertical bar z vertical bar(2), when f(tau) is a closed disk, we find that the Berezin measure instead converges to harmonic measure at z(0) relative to C\f(tau) Our results have applications to the study of the cigenvalues of random normal matrices The auxiliary results include weighted L-2-estimates for the equation partial derivative u = f when f is a suitable test function and the solution u is restricted by a polynomial growth bound at infinity.
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| 2. |
- Ameur, Yacin, et al.
(författare)
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FLUCTUATIONS OF EIGENVALUES OF RANDOM NORMAL MATRICES
- 2011
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Ingår i: Duke mathematical journal. - : Duke University Press. - 0012-7094 .- 1547-7398. ; 159:1, s. 31-81
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Tidskriftsartikel (refereegranskat)abstract
- In this article, we consider a fairly general potential in the plane and the corresponding Boltzmann-Gibbs distribution of eigenvalues of random normal matrices. As the order of the matrices tends to infinity, the eigenvalues condensate on a certain compact subset of the plane-the "droplet." We prove that fluctuations of linear statistics of eigenvalues of random normal matrices converge on compact subsets of the interior of the droplet to a Gaussian field, and we discuss various ramifications of this result.
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| 3. |
- Ameur, Yacin, et al.
(författare)
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Random normal matrices and ward identities
- 2015
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Ingår i: Annals of Probability. - 0091-1798 .- 2168-894X. ; 43:3, s. 1157-1201
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Tidskriftsartikel (refereegranskat)abstract
- We consider the random normal matrix ensemble associated with a potential in the plane of sufficient growth near infinity. It is known that asymptotically as the order of the random matrix increases indefinitely, the eigenvalues approach a certain equilibrium density, given in terms of Frostman's solution to the minimum energy problem of weighted logarithmic potential theory. At a finer scale, we may consider fluctuations of eigenvalues about the equilibrium. In the present paper, we give the correction to the expectation of the fluctuations, and we show that the potential field of the corrected fluctuations converge on smooth test functions to a Gaussian free field with free boundary conditions on the droplet associated with the potential.
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| 4. |
- Ameur, Yacin, et al.
(författare)
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Rescaling Ward Identities in the Random Normal Matrix Model
- 2019
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Ingår i: Constructive Approximation. - : Springer Science and Business Media LLC. - 0176-4276 .- 1432-0940. ; 50:1, s. 63-127
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Tidskriftsartikel (refereegranskat)abstract
- We study spacing distribution for the eigenvalues of a random normal matrix, in particular at points on the boundary of the spectrum. Our approach uses Ward’s (or the “rescaled loop”) equation—an identity satisfied by all sequential limits of the rescaled one-point functions.
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| 5. |
- Ameur, Yacin, et al.
(författare)
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Scaling limits of random normal matrix processes at singular boundary points
- 2020
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Ingår i: Journal of Functional Analysis. - : Elsevier BV. - 0022-1236. ; 278:3
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Tidskriftsartikel (refereegranskat)abstract
- We introduce a method for taking microscopic limits of normal matrix ensembles and apply it to study the behaviour near certain types of singular points on the boundary of the droplet. Our investigation includes ensembles without restrictions near the boundary, as well as hard edge ensembles, where the eigenvalues are confined to the droplet. We establish in both cases existence of new types of determinantal point fields, which differ from those which can appear at a regular boundary point, or in the bulk.
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| 6. |
- Hedenmalm, Håkan, et al.
(författare)
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Coulomb gas ensembles and Laplacian growth
- 2013
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Ingår i: Proceedings of the London Mathematical Society. - : Wiley. - 0024-6115 .- 1460-244X. ; 106:4, s. 859-907
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Tidskriftsartikel (refereegranskat)abstract
- We consider weight functions Q : C -> R that are locally in a suitable Sobolev space and impose a logarithmic growth condition from below. We use Q as a confining potential in the model of one-component plasma (2-dimensional Coulomb gas) and study the configuration of the electron cloud as the number n of electrons tends to infinity, while the confining potential is rescaled: we use mQ in place of Q and let m tend to infinity as well. We show that if m and n tend to infinity in a proportional fashion, with n/m -> t, where 0 < t <+infinity is fixed, then the electrons accumulate on a compact set S-t, which we call the droplet. The set S-t can be obtained as the coincidence set of an obstacle problem, if we remove a small set (the shallow points). Moreover, on the droplet S-t, the density of electrons is asymptotically delta Q. The growth of the droplets S-t as t increases is known as the Laplacian growth. It is well known that Laplacian growth is unstable. To analyse this feature, we introduce the notion of a local droplet, which involves removing part of the obstacle away from the set S-t. The local droplets are no longer uniquely determined by the time parameter t, but at least they may be partially ordered. We show that the growth of the local droplets may be terminated in a maximal local droplet or by the droplets' growing to infinity in some direction ('fingering').
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| 7. |
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