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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
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
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
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
The Spring Kernel: A New Paradigm for RealTime Systems
 IEEE Software
, 1991
"... Next generation realtime systems will require greater flexibility and predictability than is commonly found in today's systems. These future systems include the space station, integrated vision/robotics/AI systems, collections of humans/robots coordinating to achieve common objectives (usuall ..."
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Cited by 211 (21 self)
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Next generation realtime systems will require greater flexibility and predictability than is commonly found in today's systems. These future systems include the space station, integrated vision/robotics/AI systems, collections of humans/robots coordinating to achieve common objectives
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
Hamiltonian structure of . . .
, 1993
"... The level spacing distributions in the Gaussian Unitary Ensemble, both in the “bulk of the spectrum,” sin π(x−y) given by the Fredholm determinant of the operator with the sine kernel and on the “edge of π(x−y) the spectrum, ” given by the Airy kernel Ai(x)Ai ′ (y)−Ai(y)Ai ′ (x) (x−y), are determine ..."
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The level spacing distributions in the Gaussian Unitary Ensemble, both in the “bulk of the spectrum,” sin π(x−y) given by the Fredholm determinant of the operator with the sine kernel and on the “edge of π(x−y) the spectrum, ” given by the Airy kernel Ai(x)Ai ′ (y)−Ai(y)Ai ′ (x) (x
Analytic Variations on the Airy Distribution
, 2001
"... The Airy distribution (of the “area ” type) occurs as a limit distribution of cumulative parameters in a number of combinatorial structures, like path length in trees, area below walks, displacement in parking sequences, and it is also related to basic graph and polyomino enumeration. We obtain cur ..."
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Cited by 20 (4 self)
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probabilistic ” arguments like analytic continuation. A byproduct of this approach is the existence of relations between moments of the Airy distribution, the asymptotic expansion of the Airy function Ai(z) at +∞, and power symmetric functions of the zeros −αk of Ai(z).
Results 1  10
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248