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- Ameur, Yacin
(författare)
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A localization theorem for the planar coulomb gas in an external field
- 2021
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Ingår i: Electronic Journal of Probability. - 1083-6489. ; 26
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Tidskriftsartikel (refereegranskat)abstract
- We examine a two-dimensional Coulomb gas consisting of n identical repelling point charges at an arbitrary inverse temperature β, subjected to a suitable external field. We prove that the gas is effectively localized to a small neighbourhood of the droplet – the support of the equilibrium measure determined by the external field. More precisely, we prove that the distance between the droplet and the vacuum is with very high probability at most proportional to [Formula Presented]. This order of magnitude is known to be “tight” when β = 1 and the external field is radially symmetric. In addition, we prove estimates for the one-point function in a neighbourhood of the droplet, proving in particular a fast uniform decay as one moves beyond a distance roughly of the order [Formula Presented] from the droplet.
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| 4. |
- Ameur, Yacin
(författare)
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A note on a theorem of Sparr
- 2004
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Ingår i: Mathematica Scandinavica. - 0025-5521. ; 94:1, s. 155-160
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Tidskriftsartikel (refereegranskat)
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| 5. |
- Ameur, Yacin, et al.
(författare)
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Almost-Hermitian random matrices and bandlimited point processes
- 2023
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Ingår i: Analysis and Mathematical Physics. - 1664-2368. ; 13:3
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Tidskriftsartikel (refereegranskat)abstract
- We study the distribution of eigenvalues of almost-Hermitian random matrices associated with the classical Gaussian and Laguerre unitary ensembles. In the almost-Hermitian setting, which was pioneered by Fyodorov, Khoruzhenko and Sommers in the case of GUE, the eigenvalues are not confined to the real axis, but instead have imaginary parts which vary within a narrow “band” about the real line, of height proportional to 1/N, where N denotes the size of the matrices. We study vertical cross-sections of the 1-point density as well as microscopic scaling limits, and we compare with other results which have appeared in the literature in recent years. Our approach uses Ward’s equation and a property which we call “cross-section convergence”, which relates the large-N limit of the cross-sections of the density of eigenvalues with the equilibrium density for the corresponding Hermitian ensemble: the semi-circle law for GUE and the Marchenko–Pastur law for LUE. As an application of our approach, we prove the bulk universality of the almost-circular ensembles.
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| 6. |
- 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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| 7. |
- Ameur, Yacin, et al.
(författare)
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Beurling-Landau densities of weighted Fekete sets and correlation kernel estimates
- 2012
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Ingår i: Journal of Functional Analysis. - 0022-1236. ; 263:7, s. 1825-1861
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Tidskriftsartikel (refereegranskat)abstract
- Let Q be a suitable real function on C. An n-Fekete set corresponding to Q is a subset {z(n vertical bar) , . . . , z(nn)} of C which maximizes the expression Pi(n)(i infinity. In this note we prove that Fekete sets are, in a sense, maximally spread out with respect to the equilibrium measure. In general, our results apply only to a part of the Fekete set, which is at a certain distance away from the boundary of the droplet. However, for the potential Q = vertical bar z vertical bar(2) we obtain results which hold globally, and we conjecture that such global results are true for a wide range of potentials. (C) 2012 Elsevier Inc. All rights reserved.
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- Ameur, Yacin, et al.
(författare)
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Disk counting statistics near hard edges of random normal matrices: The multi-component regime
- 2024
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Ingår i: Advances in Mathematics. - : Elsevier BV. - 0001-8708 .- 1090-2082. ; 441
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Tidskriftsartikel (refereegranskat)abstract
- We consider a two-dimensional point process whose points are separated into two disjoint components by a hard wall, and study the multivariate moment generating function of the corresponding disk counting statistics. We investigate the “hard edge regime” where all disk boundaries are a distance of order [Formula presented] away from the hard wall, where n is the number of points. We prove that as n→+∞, the asymptotics of the moment generating function are of the form [Formula presented] and we determine the constants C1,…,C4 explicitly. The oscillatory term Fn is of order 1 and is given in terms of the Jacobi theta function. Our theorem allows us to derive various precise results on the disk counting function. For example, we prove that the asymptotic fluctuations of the number of points in one component are of order 1 and are given by an oscillatory discrete Gaussian. Furthermore, the variance of this random variable enjoys asymptotics described by the Weierstrass ℘-function.
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| 9. |
- Ameur, Yacin, et al.
(författare)
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Eigenvalues of truncated unitary matrices : disk counting statistics
- 2024
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Ingår i: Monatshefte fur Mathematik. - 0026-9255. ; 204:2, s. 197-216
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Tidskriftsartikel (refereegranskat)abstract
- Let T be an n× n truncation of an (n+ α) × (n+ α) Haar distributed unitary matrix. We consider the disk counting statistics of the eigenvalues of T. We prove that as n→ + ∞ with α fixed, the associated moment generating function enjoys asymptotics of the form exp(C1n+C2+o(1)), where the constants C1 and C2 are given in terms of the incomplete Gamma function. Our proof uses the uniform asymptotics of the incomplete Beta function.
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| 10. |
- Ameur, Yacin, et al.
(författare)
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Exponential moments for disk counting statistics at the hard edge of random normal matrices
- 2023
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Ingår i: Journal of Spectral Theory. - : European Mathematical Society - EMS - Publishing House GmbH. - 1664-039X .- 1664-0403. ; 13:3, s. 841-902
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Tidskriftsartikel (refereegranskat)abstract
- We consider the multivariate moment generating function of the disk counting statistics of a model Mittag-Leffler ensemble in the presence of a hard wall. Let n be the number of points. We focus on two regimes: (a) the “hard edge regime” where all disk boundaries are at a distance of order n1 from the hard wall, and (b) the “semi-hard edge regime” where all disk boundaries are at a distance of order √1n from the hard wall. As n → + ∞, we prove that the moment generating function enjoys asymptotics of the form (Equation presented) In both cases, we determine the constants C1;:::; C4 explicitly. We also derive precise asymptotic formulas for all joint cumulants of the disk counting function, and establish several central limit theorems. Surprisingly, and in contrast to the “bulk”, “soft edge”, and “semi-hard edge” regimes, the second and higher order cumulants of the disk counting function in the “hard edge” regime are proportional to n and not to √n.
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