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23
Fredholm Determinants, Differential Equations and Matrix Models
 COMMUNICATIONS IN MATHEMATICAL PHYSICS
, 1994
"... Orthogonal polynomial random matrix models of N x N hermitian matrices lead to Fredholm determinants of integral operators with kernel of the form (φ(x)φ(y) — ψ(x)φ(y))/x — y. This paper is concerned with the Fredholm determinants of integral operators having kernel of this form and where the und ..."
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Cited by 142 (20 self)
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Orthogonal polynomial random matrix models of N x N hermitian matrices lead to Fredholm determinants of integral operators with kernel of the form (φ(x)φ(y) — ψ(x)φ(y))/x — y. This paper is concerned with the Fredholm determinants of integral operators having kernel of this form and where the underlying set is the union of intervals J = [J ™ =1 (βiju Λ 2J) The emphasis is on the determinants thought of as functions of the endpoints a k. We show that these Fredholm determinants with kernels of the general form described above are expressible in terms of solutions of systems of PDE's as long as φ and φ satisfy a certain type of differentiation formula. The (φ, φ) pairs for the sine, Airy, and Bessel kernels satisfy such relations, as do the pairs which arise in the finite N Hermite, Laguerre and Jacobi ensembles and in matrix models of 2D quantum gravity. Therefore we shall be able to write down the systems of PDE's for these ensembles as special cases of the general system. An analysis of these equations will lead to explicit representations in terms of Painleve transcendents for the distribution functions of the largest and smallest eigenvalues in the finite N Hermite and Laguerre ensembles, and for the distribution functions of the largest and smallest singular values of rectangular matrices (of arbitrary dimensions) whose entries are independent identically distributed complex Gaussian variables. There is also an exponential variant of the kernel in which the denominator is replaced by e bx — e by, where b is an arbitrary complex number. We shall find an analogous system of differential equations in this setting. If b = i then we can interpret our operator as acting on (a subset of) the unit circle in the complex plane. As an application of this we shall write down a system of PDE's for Dyson's circular ensemble of N x N unitary matrices, and then an ODE if J is an arc of the circle.
Random matrix theory
, 2005
"... Random matrix theory is now a big subject with applications in many disciplines of science, engineering and finance. This article is a survey specifically oriented towards the needs and interests of a numerical analyst. This survey includes some original material not found anywhere else. We includ ..."
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Cited by 82 (4 self)
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Random matrix theory is now a big subject with applications in many disciplines of science, engineering and finance. This article is a survey specifically oriented towards the needs and interests of a numerical analyst. This survey includes some original material not found anywhere else. We include the important mathematics which is a very modern development, as well as the computational software that is transforming the theory into useful practice.
Application of the τfunction theory of Painlevé equations to random matrices
 PV, PIII, the LUE, JUE and CUE
, 2002
"... Okamoto has obtained a sequence of τfunctions for the PVI system expressed as a double Wronskian determinant based on a solution of the Gauss hypergeometric equation. Starting with integral solutions of the Gauss hypergeometric equation, we show that the determinant can be reexpressed as multidim ..."
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Cited by 75 (20 self)
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Okamoto has obtained a sequence of τfunctions for the PVI system expressed as a double Wronskian determinant based on a solution of the Gauss hypergeometric equation. Starting with integral solutions of the Gauss hypergeometric equation, we show that the determinant can be reexpressed as multidimensional integrals, and these in turn can be identified with averages over the eigenvalue probability density function for the Jacobi unitary ensemble (JUE), and the Cauchy unitary ensemble (CyUE) (the latter being equivalent to the circular Jacobi unitary ensemble (cJUE)). Hence these averages, which depend on four continuous parameters and the discrete parameter N, can be characterised as the solution of the second order second degree equation satisfied by the Hamiltonian in the PVI theory. We show that the Hamiltonian also satisfies an equation related to the discrete PV equation, thus providing an alternative characterisation in terms of a difference equation. In the case of the cJUE, the spectrum singularity scaled limit is considered, and the evaluation of a certain four parameter average is given in terms of the general PV transcendent in σ form. Applications are given to the evaluation of the spacing distribution for the circular unitary ensemble (CUE) and its scaled counterpart, giving formulas more succinct than those known previously; to expressions for the hard edge gap probability in the scaled Laguerre orthogonal ensemble (LOE) (parameter a a nonnegative
Increasing subsequences and the hardtosoftedge transition in matrix ensembles
 J. Phys. A
"... Our interest is in the cumulative probabilities Pr(L(t) ≤ l) for the maximum length of increasing subsequences in Poissonized ensembles of random permutations, random fixed point free involutions and reversed random fixed point free involutions. It is shown that these probabilities are equal to the ..."
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Cited by 23 (4 self)
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Our interest is in the cumulative probabilities Pr(L(t) ≤ l) for the maximum length of increasing subsequences in Poissonized ensembles of random permutations, random fixed point free involutions and reversed random fixed point free involutions. It is shown that these probabilities are equal to the hard edge gap probability for matrix ensembles with unitary, orthogonal and symplectic symmetry respectively. The gap probabilities can be written as a sum over correlations for certain determinantal point processes. From these expressions a proof can be given that the limiting form of Pr(L(t) ≤ l) in the three cases is equal to the soft edge gap probability for matrix ensembles with unitary, orthogonal and symplectic symmetry respectively, thereby reclaiming theorems due to BaikDeiftJohansson and BaikRains. 1
Universality for orthogonal and symplectic Laguerretype ensembles
 J. Statist. Phys
, 2007
"... Abstract. We give a proof of the Universality Conjecture for orthogonal (β = 1) and symplectic (β = 4) random matrix ensembles of Laguerretype in the bulk of the spectrum as well as at the hard and soft spectral edges. Our results are stated precisely in the Introduction (Theorems 1.1, 1.4, 1.6 and ..."
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Cited by 16 (0 self)
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Abstract. We give a proof of the Universality Conjecture for orthogonal (β = 1) and symplectic (β = 4) random matrix ensembles of Laguerretype in the bulk of the spectrum as well as at the hard and soft spectral edges. Our results are stated precisely in the Introduction (Theorems 1.1, 1.4, 1.6 and Corollaries 1.2, 1.5, 1.7). They concern the appropriately rescaled kernels Kn,β, correlation and cluster functions, gap probabilities and the distributions of the largest and smallest eigenvalues. Corresponding results for unitary (β = 2) Laguerretype ensembles have been proved by the fourth author in [23]. The varying weight case at the hard spectral edge
τFUNCTION EVALUATION OF GAP PROBABILITIES IN ORTHOGONAL AND SYMPLECTIC MATRIX ENSEMBLES
, 2002
"... It has recently been emphasized that all known exact evaluations of gap probabilities for classical unitary matrix ensembles are in fact τfunctions for certain Painlevé systems. We show that all exact evaluations of gap probabilities for classical orthogonal matrix ensembles, either known or deriva ..."
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Cited by 12 (4 self)
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It has recently been emphasized that all known exact evaluations of gap probabilities for classical unitary matrix ensembles are in fact τfunctions for certain Painlevé systems. We show that all exact evaluations of gap probabilities for classical orthogonal matrix ensembles, either known or derivable from the existing literature, are likewise τfunctions for certain Painlevé systems. In the case of symplectic matrix ensembles all exact evaluations, either known or derivable from the existing literature, are identified as the mean of two τfunctions, both of which correspond to Hamiltonians satisfying the same differential equation, differing only in the boundary condition. Furthermore the product of these two τfunctions gives the gap probability in the corresponding unitary symmetry case, while one of those τfunctions is the gap probability in the corresponding orthogonal symmetry case. 1
On the largest singular values of random matrices with independent Cauchy entries
 J. MATH. PHYS
, 2005
"... We apply the method of determinants to study the distribution of the largest singular values of large m × n real rectangular random matrices with independent Cauchy entries. We show that statistical properties of the (rescaled by a factor 1 m 2 n 2) largest singular values agree in the limit with th ..."
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Cited by 7 (0 self)
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We apply the method of determinants to study the distribution of the largest singular values of large m × n real rectangular random matrices with independent Cauchy entries. We show that statistical properties of the (rescaled by a factor 1 m 2 n 2) largest singular values agree in the limit with the statistics of the Poisson random point process with the intensity 1 π x−3/2 and, therefore, are different from the TracyWidom law. Among other corollaries of our method we show an interesting connection between the mathematical expectations of the determinants of the complex rectangular m×n standard Wishart ensemble and the real rectangular 2m×2n standard Wishart ensemble.
2006, Relationships between τfunctions and Fredholm determinant expressions for gap probabilities in random matrix theory, Nonlinearity 19
"... Abstract. The gap probabilities at the hard and soft edges of scaled random matrix ensembles with orthogonal symmetry are known in terms of τfunctions. Extending recent work relating to the soft edge, it is shown that these τfunctions, and their generalizations to contain a generating function lik ..."
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Cited by 5 (1 self)
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Abstract. The gap probabilities at the hard and soft edges of scaled random matrix ensembles with orthogonal symmetry are known in terms of τfunctions. Extending recent work relating to the soft edge, it is shown that these τfunctions, and their generalizations to contain a generating function like parameter, can be expressed as Fredholm determinants. These same Fredholm determinants also occur in exact expressions for gap probabilities in scaled random matrix ensembles with unitary and symplectic symmetry. 1.