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Universality at the edge of the spectrum for unitary, orthogonal and symplectic ensembles of random matrices
 Comm. Pure Appl. Math
"... Abstract. We prove universality at the edge of the spectrum for unitary (β = 2), orthogonal (β = 1) and symplectic (β = 4) ensembles of random matrices in the scaling limit for a class of weights w(x) = e −V (x) where V is a polynomial, V (x) = κ2mx 2m + · · · , κ2m> 0. The precise statement ..."
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Cited by 55 (6 self)
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Abstract. We prove universality at the edge of the spectrum for unitary (β = 2), orthogonal (β = 1) and symplectic (β = 4) ensembles of random matrices in the scaling limit for a class of weights w(x) = e −V (x) where V is a polynomial, V (x) = κ2mx 2m + · · · , κ2m> 0. The precise statement of our results is given in Theorem 1.1 and Corollaries 1.2, 1.3 below. For a proof of universality in the bulk of the spectrum, for the same class of weights, for unitary ensembles see [DKMVZ2], and for orthogonal and symplectic ensembles see [DG]. Our starting point in the unitary case is [DKMVZ2], and for the orthogonal and symplectic cases we rely on our recent work [DG], which in turn depends on the earlier work of Widom [W] and Tracy and Widom [TW2]. As in [DG], the uniform Plancherel–Rotach type asymptotics for the orthogonal polynomials found in [DKMVZ2] plays a central role. The formulae in [W] express the correlation kernels for β = 1 and 4 as a sum of a Christoffel–Darboux (CD) term, as in the case β = 2, together with a correction term. In the bulk scaling limit [DG], the correction term is of lower order and does not contribute to the limiting form of the correlation kernel. By contrast, in the edge scaling limit considered here, the CD term and the correction term contribute to the same order: this leads to additional technical difficulties over and above [DG]. 1.
Symmetry of matrixvalued stochastic processes and noncolliding diffusion particle systems
 J. Math. Phys
, 2004
"... As an extension of the theory of Dyson’s Brownian motion models for the standard Gaussian randommatrix ensembles, we report a systematic study of hermitian matrixvalued processes and their eigenvalue processes associated with the chiral and nonstandard randommatrix ensembles. In addition to the n ..."
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Cited by 45 (19 self)
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As an extension of the theory of Dyson’s Brownian motion models for the standard Gaussian randommatrix ensembles, we report a systematic study of hermitian matrixvalued processes and their eigenvalue processes associated with the chiral and nonstandard randommatrix ensembles. In addition to the noncolliding Brownian motions, we introduce a oneparameter family of temporally homogeneous noncolliding systems of the Bessel processes and a twoparameter family of temporally inhomogeneous noncolliding systems of Yor’s generalized meanders and show that all of the ten classes of eigenvalue statistics in the AltlandZirnbauer classification are realized as particle distributions in the special cases of these diffusion particle systems. As a corollary of each equivalence in distribution of a temporally inhomogeneous eigenvalue process and a noncolliding diffusion process, a stochasticcalculus proof of a version of the HarishChandra (ItzyksonZuber) formula of integral over unitary group is established. I
On the relations between orthogonal, symplectic and unitary matrix models. J.Stat.Phys
, 1999
"... For the unitary ensembles of N × N Hermitian matrices associated with a weight function w there is a kernel, expressible in terms of the polynomials orthogonal with respect to the weight function, which plays an important role. For the orthogonal and symplectic ensembles of Hermitian matrices there ..."
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Cited by 45 (2 self)
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For the unitary ensembles of N × N Hermitian matrices associated with a weight function w there is a kernel, expressible in terms of the polynomials orthogonal with respect to the weight function, which plays an important role. For the orthogonal and symplectic ensembles of Hermitian matrices there are 2 × 2 matrix kernels, usually constructed using skeworthogonal polynomials, which play an analogous role. These matrix kernels are determined by their upper lefthand entries. We derive formulas expressing these entries in terms of the scalar kernel for the corresponding unitary ensembles. We also show that whenever w ′ /w is a rational function the entries are equal to the scalar kernel plus some extra terms whose number equals the order of w ′ /w. General formulas are obtained for these extra terms. We do not use skeworthogonal polynomials in the derivations. 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
Infinite systems of noncolliding generalized meanders and RiemannLiouville differintegrals
, 2005
"... Yor’s generalized meander is a temporally inhomogeneous modification of the 2(ν+1)dimensional Bessel process with ν> −1, in which the inhomogeneity is indexed by κ ∈ [0, 2(ν + 1)). We introduce the noncolliding particle systems of the generalized meanders and prove that they are the Pfaffian p ..."
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Cited by 16 (12 self)
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Yor’s generalized meander is a temporally inhomogeneous modification of the 2(ν+1)dimensional Bessel process with ν> −1, in which the inhomogeneity is indexed by κ ∈ [0, 2(ν + 1)). We introduce the noncolliding particle systems of the generalized meanders and prove that they are the Pfaffian processes, in the sense that any multitime correlation function is given by a Pfaffian. In the infinite particle limit, we show that the elements of matrix kernels of the obtained infinite Pfaffian processes are generally expressed by the RiemannLiouville differintegrals of functions comprising the Bessel functions Jν used in the fractional calculus, where orders of differintegration are determined by ν − κ. As special cases of the two parameters (ν, κ), the present infinite systems include the quaternion determinantal processes studied by Forrester, Nagao and Honner and by Nagao, which exhibit the temporal transitions between the universality classes of random matrix theory.
On Orthogonal and Symplectic Matrix Ensembles Associated with a Class of Weight Functions
, 1998
"... For the unitary ensembles of N × N Hermitian matrices associated with a weight function w there is a kernel, expressible in terms of the polynomials orthogonal with respect to the weight function, which plays an important role. For example the npoint correlation function and the spacing probabiliti ..."
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For the unitary ensembles of N × N Hermitian matrices associated with a weight function w there is a kernel, expressible in terms of the polynomials orthogonal with respect to the weight function, which plays an important role. For example the npoint correlation function and the spacing probabilities have nice representations in terms of this kernel. For the orthogonal and symplectic ensembles of Hermitian matrices there are 2×2 matrix kernels, usually constructed using skeworthogonal polynomials, which play an analogous role. These matrix kernels are determined by their upper lefthand entries. We show here that whenever w ′ /w is a rational function these entries are equal to the scalar kernel for the corresponding unitary ensemble (but with w replaced by w 2 for orthogonal ensembles and N replaced by 2N for symplectic ensembles) plus some “extra ” terms whose number equals the order of w ′ /w. General formulas are obtained for these extra terms. We do not use skeworthogonal polynomials in the derivation. 1.
On the Microscopic Spectra of the Massive Dirac Operator for Chiral Orthogonal and
, 2000
"... Using Random Matrix Theory we set out to compute the microscopic correlators of the Euclidean Dirac operator in four dimensions. In particular we consider: the chiral Orthogonal Ensemble (χOE), corresponding to a YangMills theory with two colors and fermions in the fundamental representation, and t ..."
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Using Random Matrix Theory we set out to compute the microscopic correlators of the Euclidean Dirac operator in four dimensions. In particular we consider: the chiral Orthogonal Ensemble (χOE), corresponding to a YangMills theory with two colors and fermions in the fundamental representation, and the chiral Symplectic Ensemble (χSE), corresponding to any number of colors and fermions in the adjoint representation. In both cases we deal with an arbitrary number of massive fermions. We use a recent method proposed by H. Widom for deriving closed formulas for the scalar kernels from which all spectral correlation functions of the χGOE and χGSE can be determined. Moreover, we obtain complete analytic expressions of such correlators in the double microscopic limit, extending previously known results of fourdimensional QCD at β = 1 and β = 4 to the One of the most successful and wellestablished physical applications of Random Matrix Theory (RMT) is the analysis of Quantum Chromodynamics (QCD) at low energies. In particular, the spectral statistical properties of the Euclidean Dirac operator in the infrared