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The Spectrum of Coupled Random Matrices
, 1999
"... this paper, we will use the following operators e ..."
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Cited by 37 (10 self)
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this paper, we will use the following operators e
Introduction to random matrices
 the proceedings of the 8 th Scheveningen Conference, Springer Lecture Notes in Physics
, 1993
"... ..."
Asymptotics for the Fredholm determinant of the sine kernel on a union of intervals
 Commun. Math. Phys
, 1995
"... The sine kernel ..."
Gap probability in the spectrum of random matrices and asymptotics of polynomials orthogonal on an arc of the unit circle.
, 2004
"... ..."
ON THE NUMERICAL EVALUATION OF FREDHOLM DETERMINANTS
, 804
"... Abstract. Some significant quantities in mathematics and physics are most naturally expressed as the Fredholm determinant of an integral operator, most notably many of the distribution functions in random matrix theory. Though their numerical values are of interest, there is no systematic numerical ..."
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Cited by 10 (5 self)
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Abstract. Some significant quantities in mathematics and physics are most naturally expressed as the Fredholm determinant of an integral operator, most notably many of the distribution functions in random matrix theory. Though their numerical values are of interest, there is no systematic numerical treatment of Fredholm determinants to be found in the literature. Instead, the few numerical evaluations that are available rely on eigenfunction expansions of the operator, if expressible in terms of special functions, or on alternative, numerically more straightforwardly accessible analytic expressions, e.g., in terms of Painlevé transcendents, that have masterfully been derived in some cases. In this paper we close the gap in the literature by studying projection methods and, above all, a simple, easily implementable, general method for the numerical evaluation of Fredholm determinants that is derived from the classical Nyström method for the solution of Fredholm equations of the second kind. Using Gauss–Legendre or Clenshaw– Curtis as the underlying quadrature rule, we prove that the approximation error essentially behaves like the quadrature error for the sections of the kernel. In particular, we get exponential convergence for analytic kernels, which are typical in random matrix theory. The application of the method to the distribution functions of the Gaussian unitary ensemble (GUE), in the bulk and the edge scaling limit, is discussed in detail. After extending the method to systems of integral operators, we evaluate the twopoint correlation functions of the more recently studied Airy and Airy 1 processes. Key words. Fredholm determinant, Nyström’s method, projection method, trace class operators, random
Asymptotics of TracyWidom distributions and the total integral of a Painlevé
 II function, Comm. Math. Phys
"... The TracyWidom distribution functions involve integrals of a Painlevé II function starting from positive infinity. In this paper, we express the TracyWidom distribution functions in terms of integrals starting from minus infinity. There are two consequences of these new representations. The first ..."
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Cited by 7 (1 self)
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The TracyWidom distribution functions involve integrals of a Painlevé II function starting from positive infinity. In this paper, we express the TracyWidom distribution functions in terms of integrals starting from minus infinity. There are two consequences of these new representations. The first is the evaluation of the total integral of the HastingsMcLeod solution of the Painlevé II equation. The second is the evaluation of the constant term of the asymptotic expansions of the TracyWidom distribution functions as the distribution parameter approaches minus infinity. For the GUE TracyWidom distribution function, this gives an alternative proof of the recent work of Deift, Its, and Krasovsky. The constant terms for the GOE and GSE TracyWidom distribution functions are new.
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 4 (3 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.
Asymptotics of a Class of Fredholm Determinants
, 1998
"... In this expository article we describe the asymptotics of certain Fredholm determinants which provide solutions to the cylindrical Toda equations, and we explain how these asymptotics are derived. The connection with Fredholm determinants arising in the theory of random matrices, and their asymptoti ..."
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Cited by 1 (1 self)
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In this expository article we describe the asymptotics of certain Fredholm determinants which provide solutions to the cylindrical Toda equations, and we explain how these asymptotics are derived. The connection with Fredholm determinants arising in the theory of random matrices, and their asymptotics, are also discussed. 1.
Some computable WienerHopf determinants and polynomials orthogonal on an arc of the unit circle
"... Some WienerHopf determinants on [0, s] are calculated explicitly for all s > 0. Their symbols are zero on an interval and they are related to the determinant with the sinekernel appearing in the random matrix theory. The determinants are calculated by taking limits of Toeplitz determinants, which ..."
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Some WienerHopf determinants on [0, s] are calculated explicitly for all s > 0. Their symbols are zero on an interval and they are related to the determinant with the sinekernel appearing in the random matrix theory. The determinants are calculated by taking limits of Toeplitz determinants, which in turn are found from the related systems of polynomials orthogonal on an arc of the unit circle. As is known, the latter polynomials are connected to those orthogonal on an interval of the real axis. This connection is somewhat extended here. The determinants we compute originate from the BernsteinSzegö (in particular Chebyshev) orthogonal polynomials.