| 1. |
- 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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| 2. |
- 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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| 3. |
- Ameur, Yacin, et al.
(författare)
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Szegő Type Asymptotics for the Reproducing Kernel in Spaces of Full-Plane Weighted Polynomials
- 2023
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Ingår i: Communications in Mathematical Physics. - : Springer Science and Business Media LLC. - 0010-3616 .- 1432-0916. ; 398:3, s. 1291-1348
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Tidskriftsartikel (refereegranskat)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 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.
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| 4. |
- Turova, Tatyana, et al.
(författare)
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Particle Dynamics in Time-Dependent Coulomb Field
- 2021
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Ingår i: Markov Processes and Related Fields. - 1024-2953. ; 27:2, s. 197-227
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Tidskriftsartikel (refereegranskat)abstract
- We study a non-autonomous differential equation describing the one-dimensional motion of a particle interacting with a charged source at the origin. The charge of the source is piece-wise constant; it alternates between two different values, one being positive and the other being negative. We show that the alternation of the charge enriches the behaviour of the system, exhibiting qualitatively new features.In particular, we find necessary and sufficient conditions for the parameters of the model which yield periodic bounded solutions, and we describe some necessary conditions as well.
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