| 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. |
- Berntson, Björn K., et al.
(författare)
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Elliptic soliton solutions of the spin non-chiral intermediate long-wave equation
- 2023
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Ingår i: Letters in Mathematical Physics. - : Springer Nature. - 0377-9017 .- 1573-0530. ; 113:3
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Tidskriftsartikel (refereegranskat)abstract
- We construct elliptic multi-soliton solutions of the spin non-chiral intermediate long-wave (sncILW) equation with periodic boundary conditions. These solutions are obtained by a spin-pole ansatz including a dynamical background term; we show that this ansatz solves the periodic sncILW equation provided the spins and poles satisfy the elliptic A-type spin Calogero-Moser (sCM) system with certain constraints on the initial conditions. The key to this result is a Backlund transformation for the elliptic sCM system which includes a non-trivial dynamical background term. We also present solutions of the sncILW equation on the real line and of the spin Benjamin-Ono equation which generalize previously obtained solutions by allowing for a non-trivial background term.
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| 4. |
- Berntson, Bjorn K., et al.
(författare)
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Nonchiral intermediate long-wave equation and interedge effects in narrow quantum Hall systems
- 2020
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Ingår i: Physical Review B. - : AMER PHYSICAL SOC. - 2469-9950 .- 2469-9969. ; 102:15
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Tidskriftsartikel (refereegranskat)abstract
- We present a nonchiral version of the intermediate long-wave (ILW) equation that can model nonlinear waves propagating on two opposite edges of a quantum Hall system, taking into account interedge interactions. We obtain exact soliton solutions governed by the hyperbolic Calogero-Moser-Sutherland (CMS) model, and we give a Lax pair, a Hirota form, and conservation laws for this new equation. We also present a periodic nonchiral ILW equation, together with its soliton solutions governed by the elliptic CMS model.
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| 5. |
- Berntson, Bjorn K., et al.
(författare)
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On the non-chiral intermediate long wave equation : II. Periodic case
- 2022
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Ingår i: Nonlinearity. - : IOP Publishing. - 0951-7715 .- 1361-6544. ; 35:8, s. 4517-4548
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Tidskriftsartikel (refereegranskat)abstract
- We study integrability properties of the non-chiral intermediate long wave (ncILW) equation with periodic boundary conditions. The ncILW equation was recently introduced by the authors as a parity-invariant relative of the intermediate long wave equation. For this new equation we: (a) derive a Lax pair, (b) derive a Hirota bilinear form,
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| 6. |
- Berntson, Bjorn K., et al.
(författare)
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On the non-chiral intermediate long wave equation
- 2022
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Ingår i: Nonlinearity. - : IOP Publishing. - 0951-7715 .- 1361-6544. ; 35:8, s. 4549-4584
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Tidskriftsartikel (refereegranskat)abstract
- We study integrability properties of the non-chiral intermediate long wave equation recently introduced by the authors as a parity-invariant variant of the intermediate long wave equation. For this new equation we: (a) derive a Lax pair, (b) derive a Hirota bilinear form,
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| 7. |
- Berntson, Björn K., et al.
(författare)
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Spin generalizations of the Benjamin-Ono equation
- 2022
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Ingår i: Letters in Mathematical Physics. - : Springer Nature. - 0377-9017 .- 1573-0530. ; 112:3
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Tidskriftsartikel (refereegranskat)abstract
- We present new soliton equations related to the A-type spin Calogero-Moser (CM) systems introduced by Gibbons and Hermsen. These equations are spin generalizations of the Benjamin-Ono (BO) equation and the recently introduced non-chiral intermediate long-wave (ncILW) equation. We obtain multi-soliton solutions of these spin generalizations of the BO equation and the ncILW equation via a spin-pole ansatz where the spin-pole dynamics is governed by the spin CM system in the rational and hyperbolic cases, respectively. We also propose physics applications of the new equations, and we introduce a spin generalization of the standard intermediate long-wave equation which interpolates between the matrix Korteweg-de Vries equation, the Heisenberg ferromagnet equation, and the spin BO equation.
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| 8. |
- 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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| 9. |
- 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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| 10. |
- 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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