| 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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Eigenvalues of truncated unitary matrices : disk counting statistics
- 2023
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Ingår i: Monatshefte fur Mathematik. - 0026-9255.
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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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| 3. |
- 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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| 4. |
- Atkin, Max R., et al.
(författare)
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On the ratio probability of the smallest eigenvalues in the Laguerre unitary ensemble
- 2018
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Ingår i: Nonlinearity. - : IOP PUBLISHING LTD. - 0951-7715 .- 1361-6544. ; 31:4, s. 1155-1196
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Tidskriftsartikel (refereegranskat)abstract
- We study the probability distribution of the ratio between the second smallest and smallest eigenvalue in the n x n Laguerre unitary ensemble. The probability that this ratio is greater than r > 1 is expressed in terms of an n x n Hankel determinant with a perturbed Laguerre weight. The limiting probability distribution for the ratio as n -> infinity is found as an integral over (0, infinity) containing two functions q(1)(x) and q(2)(x). These functions satisfy a system of two coupled Painleve V equations, which are derived from a Lax pair of a Riemann-Hilbert problem. We compute asymptotic behaviours of these functions as rx -> 0(+) and (r - 1)x -> infinity, as well as large n asymptotics for the associated Hankel determinants in several regimes of r and x.
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| 5. |
- Blackstone, Elliot, et al.
(författare)
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Gap probabilities in the bulk of the Airy process
- 2022
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Ingår i: Random Matrices. Theory and Applications. - : World Scientific Pub Co Pte Ltd. - 2010-3263. ; 11:02
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Tidskriftsartikel (refereegranskat)abstract
- We consider the probability that no points lie on g large intervals in the bulk of the Airy point process. We make a conjecture for all the terms in the asymptotics up to and including the oscillations of order 1, and we prove this conjecture for g = 1.
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| 6. |
- Blackstone, Elliot, et al.
(författare)
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Oscillatory Asymptotics for the Airy Kernel Determinant on Two Intervals
- 2022
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Ingår i: International mathematics research notices. - : Oxford University Press (OUP). - 1073-7928 .- 1687-0247. ; 2022:4, s. 2636-2687
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Tidskriftsartikel (refereegranskat)abstract
- We obtain asymptotics for the Airy kernel Fredholm determinant on two intervals. We give explicit formulas for all the terms up to and including the oscillations of order 1, which are expressed in terms of Jacobi theta-functions.
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| 7. |
- Blackstone, Elliot, et al.
(författare)
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The Bessel kernel determinant on large intervals and Birkhoff's ergodic theorem
- 2023
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Ingår i: Communications on Pure and Applied Mathematics. - : Wiley. - 0010-3640 .- 1097-0312. ; 76:11, s. 3300-3345
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Tidskriftsartikel (refereegranskat)abstract
- The Bessel process models the local eigenvalue statistics near 0 of certain large positive definite matrices. In this work, we consider the probability (Figure presented.) where (Figure presented.) and (Figure presented.) is any non-negative integer. We obtain asymptotics for this probability as the size of the intervals becomes large, up to and including the oscillations of order 1. In these asymptotics, the most intricate term is a one-dimensional integral along a linear flow on a g-dimensional torus, whose integrand involves ratios of Riemann θ-functions associated to a genus g Riemann surface. We simplify this integral in two generic cases: (a) If the flow is ergodic, we compute the leading term in the asymptotics of this integral explicitly using Birkhoff's ergodic theorem. (b) If the linear flow has certain “good Diophantine properties”, we obtain improved estimates on the error term in the asymptotics of this integral. In the case when the flow is both ergodic and has “good Diophantine properties” (which is always the case for (Figure presented.), and “almost always” the case for (Figure presented.)), these results can be combined, yielding particularly precise and explicit large gap asymptotics.
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| 8. |
- Blackstone, Elliot, et al.
(författare)
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Toeplitz determinants with a one-cut regular potential and Fisher-Hartwig singularities I. Equilibrium measure supported on the unit circle
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Ingår i: Proceedings of the Royal Society of Edinburgh Section A: Mathematics. - 0308-2105.
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Tidskriftsartikel (refereegranskat)abstract
- We consider Toeplitz determinants whose symbol has: (i) a one-cut regular potential, (ii) Fisher-Hartwig singularities and (iii) a smooth function in the background. The potential is associated with an equilibrium measure that is assumed to be supported on the whole unit circle. For constant potentials, the equilibrium measure is the uniform measure on the unit circle and our formulas reduce to well-known results for Toeplitz determinants with Fisher-Hartwig singularities. For non-constant, our results appear to be new even in the case of no Fisher-Hartwig singularities. As applications of our results, we derive various statistical properties of a determinantal point process which generalizes the circular unitary ensemble.
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| 9. |
- Blackstone, Elliot, et al.
(författare)
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Toeplitz determinants with a one-cut regular potential and Fisher-Hartwig singularities I. Equilibrium measure supported on the unit circle
- 2023
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Ingår i: Proceedings of the Royal Society of Edinburgh. Section A Mathematics. - : Cambridge University Press (CUP). - 0308-2105 .- 1473-7124.
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Tidskriftsartikel (refereegranskat)abstract
- We consider Toeplitz determinants whose symbol has: (i) a one-cut regular potential, (ii) Fisher-Hartwig singularities and (iii) a smooth function in the background. The potential is associated with an equilibrium measure that is assumed to be supported on the whole unit circle. For constant potentials, the equilibrium measure is the uniform measure on the unit circle and our formulas reduce to well-known results for Toeplitz determinants with Fisher-Hartwig singularities. For non-constant, our results appear to be new even in the case of no Fisher-Hartwig singularities. As applications of our results, we derive various statistical properties of a determinantal point process which generalizes the circular unitary ensemble.
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| 10. |
- Byun, Sung Soo, et al.
(författare)
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On the almost-circular symplectic induced Ginibre ensemble
- 2023
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Ingår i: Studies in Applied Mathematics. - : Wiley. - 0022-2526 .- 1467-9590. ; 150:1, s. 184-217
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Tidskriftsartikel (refereegranskat)abstract
- We consider the symplectic-induced Ginibre process, which is a Pfaffian point process on the plane. Let N be the number of points. We focus on the almost-circular regime where most of the points lie in a thin annulus (Formula presented.) of width (Formula presented.) as (Formula presented.). Our main results are the bulk scaling limits of all correlation functions near the real axis, and also away from the real axis. Near the real axis, the limiting correlation functions are Pfaffians with a new correlation kernel, which interpolates the limiting kernels in the bulk of the symplectic Ginibre ensemble and of the antisymmetric Gaussian Hermitian ensemble of odd size. Away from the real axis, the limiting correlation functions are determinants, and the kernel is the same as the one appearing in the bulk limit of almost-Hermitian random matrices. Furthermore, we obtain precise large N asymptotics for the probability that no points lie outside (Formula presented.), as well as of several other “semi-large” gap probabilities.
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