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LevelSpacing Distributions and the Airy Kernel
 COMMUNICATIONS IN MATHEMATICAL PHYSICS
, 1994
"... Scaling levelspacing distribution functions in the "bulk of the spectrum" in random matrix models of N x N hermitian matrices and then going to the limit N — » oo leads to the Fredholm determinant of the sine kernel sinπ(x — y)/π(x — y). Similarly a scaling limit at the "edge o ..."
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Cited by 430 (24 self)
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;quot;edge of the spectrum " leads to the Airy kernel [Ai(x) Ai(y) — Ai (x) Ai(y)]/(x — y). In this paper we derive analogues for this Airy kernel of the following properties of the sine kernel: the completely integrable system of P.D.E.'s found by Jimbo, Miwa, Mori, and Sato; the expression, in the case of a
Asymptotics of the Airykernel determinant
, 2006
"... The authors use RiemannHilbert methods to compute the constant that arises in the asymptotic behavior of the Airykernel determinant of random matrix theory. ..."
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Cited by 12 (4 self)
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The authors use RiemannHilbert methods to compute the constant that arises in the asymptotic behavior of the Airykernel determinant of random matrix theory.
Airy kernel and Painlevé II
 In Isomonodromic deformations and applications in physics, volume 31 of CRM Proceedings & Lecture
"... We prove that the distribution function of the largest eigenvalue in the Gaussian Unitary Ensemble (GUE) in the edge scaling limit is expressible in terms of Painlevé II. Our goal is to concentrate on this important example of the connection between random matrix theory and integrable systems, and i ..."
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Cited by 4 (0 self)
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We prove that the distribution function of the largest eigenvalue in the Gaussian Unitary Ensemble (GUE) in the edge scaling limit is expressible in terms of Painlevé II. Our goal is to concentrate on this important example of the connection between random matrix theory and integrable systems, and in so doing to introduce the newcomer to the subject as a whole. We also give sketches of the results for the limiting distribution of the largest eigenvalue in the Gaussian Orthogonal Ensemble (GOE) and the Gaussian Symplectic Ensemble (GSE). This work we did some years ago in a more general setting. These notes, therefore, are not meant for experts in the field. 1
LevelSpacing Distributions and the Airy Kernel
, 1992
"... Scaling levelspacing distribution functions in the “bulk of the spectrum ” in random matrix models of N × N hermitian matrices and then going to the limit N → ∞, leads to the Fredholm determinant of the sine kernel sinπ(x − y)/π(x − y). Similarly a scaling limit at the “edge of the spectrum ” leads ..."
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” leads to the Airy kernel [Ai(x)Ai′(y) − Ai ′(x)Ai(y)] /(x − y). In this paper we derive analogues for this Airy kernel of the following properties of the sine kernel: the completely integrable system of P.D.E.’s found by Jimbo, Miwa, Môri and Sato; the expression, in the case of a single interval
ITD 92/93–11 LevelSpacing Distributions and the Airy Kernel
, 1992
"... Scaling levelspacing distribution functions in the “bulk of the spectrum ” in random matrix models of N ×N hermitian matrices and then going to the limit N → ∞, leads to the Fredholm determinant of the sine kernel sin π(x − y)/π(x − y). Similarly a double scaling limit at the “edge of the spectrum ..."
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” leads to the Airy kernel [Ai(x)Ai′(y) − Ai ′(x)Ai(y)] /(x−y). We announce analogies for this Airy kernel of the following properties of the sine kernel: the completely integrable system of P.D.E.’s found by Jimbo, Miwa, Môri and Sato; the expression, in the case of a single interval, of the Fredholm
ON THE AIRY REPRODUCING KERNEL, SAMPLING SERIES, AND QUADRATURE FORMULA
"... Abstract. We determine the class of entire functions for which the Airy kernel (of random matrix theory) is a reproducing kernel. We deduce an Airy sampling series and quadrature formula. Our results are analogues of well known ones for the Bessel kernel. The need for these arises in investigating u ..."
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Cited by 2 (0 self)
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Abstract. We determine the class of entire functions for which the Airy kernel (of random matrix theory) is a reproducing kernel. We deduce an Airy sampling series and quadrature formula. Our results are analogues of well known ones for the Bessel kernel. The need for these arises in investigating
The arctic circle boundary and the Airy process
 Ann. Prob
, 2005
"... Abstract. We prove that the, appropriately rescaled, boundary of the north polar region in the Aztec diamond converges to the Airy process. The proof uses certain determinantal point processes given by the extended Krawtchouk kernel. We also prove a version of Propp’s conjecture concerning the struc ..."
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Cited by 88 (6 self)
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Abstract. We prove that the, appropriately rescaled, boundary of the north polar region in the Aztec diamond converges to the Airy process. The proof uses certain determinantal point processes given by the extended Krawtchouk kernel. We also prove a version of Propp’s conjecture concerning
Zeros of Airy Function and Relaxation Process
, 2009
"... Onedimensional system of Brownian motions called Dyson’s model is the particle system with longrange repulsive forces acting between any pair of particles, where the strength of force is β/2 times the inverse of particle distance. When β = 2, it is realized as the Brownian motions in one dimension ..."
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Cited by 21 (12 self)
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continuous kernel. The Airy function Ai(z) is an entire function with zeros all located on the negative part of the real axis R. We consider Dyson’s model with β = 2 starting from the first N zeros of Ai(z), 0> a1> · · ·> aN, N ≥ 2. In order to properly control the effect of such initial
Airy Functions for Compact Lie Groups.
"... Abstract. The classical Airy function has been generalised by Kontsevich to a function of a matrix argument, which is an integral over the space of (skew) hermitian matrices of a unitaryinvariant exponential kernel. In this paper, the Kontsevich integral is generalised to integrals over the Lie al ..."
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Abstract. The classical Airy function has been generalised by Kontsevich to a function of a matrix argument, which is an integral over the space of (skew) hermitian matrices of a unitaryinvariant exponential kernel. In this paper, the Kontsevich integral is generalised to integrals over the Lie
Results 1  10
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53