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## Universality for orthogonal and symplectic Laguerre-type ensembles (2007)

Venue: | J. Statist. Phys |

Citations: | 16 - 0 self |

### Citations

982 |
Aspects of Multivariate Statistical Theory
- Muirhead
- 1982
(Show Context)
Citation Context ...entries that are independently distributed standard Gaussian variables, have an eigenvalue probability density function of the form (1.2), (1.3) with β = 1, γ = (N − n − 1)/2 and Q(x) = x/2 (see e.g. =-=[15]-=-). In physics, Laguerre ensembles emerge e.g. in the study of Dirac operators in quantum chromodynamics and in the study of disordered superconductors in mesoscopic physics, see e.g. [4, 24]. Here we ... |

423 |
Trace Ideals and Their Applications
- Simon
- 2005
(Show Context)
Citation Context ...ant det2 is defined by det2(I + A) ≡ det ( (I + A)e −A) e tr(A11+A22) for 2×2 block operators A = (Aij)i,j=1,2 with A11, A22 in trace class and A12, A21 Hilbert–Schmidt (cf. [8, below Corollary 1.2], =-=[19]-=-). Define ( 1 ∆n(ξ, η) := g(ξ) ν2 K n (νn) n,1 (˜ ξ (n) , ˜η (n) ) − K (1) (ξ, η) In order to prove the convergence of (6.33) to √ ( ) 1 − det2 I − g(ξ)K (1)(ξ, η)g(η) −1 |L2((0,s]) 2 , ) g(η) −1 . it... |

190 |
The isomonodromy approach to matrix models in 2D quantum gravity.
- Fokas, s, et al.
- 1992
(Show Context)
Citation Context ...ERBAUER, AND VANLESSEN In order to obtain the asymptotics of the functions ˆ ψr (r = 1, 2), see (4.1), we write them in terms of the RH problem for orthogonal polynomials due to Fokas, Its and Kitaev =-=[9]-=-. Let Y be the solution of the RH problem for orthogonal polynomials associated to the weight x α e −V (x) on [0, ∞), Y (z) = ( 1 γn pn(z) 1 γn C(pnw)(z) ) , for z ∈ C \ [0, ∞), −2πiγn−1pn−1(z) −2πiγn... |

169 |
Random Matrices, 2nd ed.,
- Mehta
- 1991
(Show Context)
Citation Context ...invariant distributions of Laguerre type (1.1) dPn,β(M) = Pn,β(M)dM = 1 Zn,β det(Wγ(M))e −tr Q(M) dM, for β = 1, 2 and 4, the so-called Orthogonal, Unitary and Symplectic ensembles, respectively (see =-=[14]-=-). For β = 1, 2, 4, the ensemble consists of n × n real symmetric matrices, n × n Hermitian matrices, and 2n ×2n Hermitian self-dual matrices (see [14]), respectively. The above terminology for β = 1,... |

160 |
The spectrum edge of random matrix ensembles.
- Forrester
- 1993
(Show Context)
Citation Context ...onsidered 1 . Consequently, the limiting local eigenvalue statistics agree for all ensembles (1.1) with the corresponding limiting statistics in the well studied classical cases of linear Q (see e.g. =-=[17, 10, 21, 16, 11]-=- and references therein). Ensembles (1.1) with linear Q are called Laguerre ensembles because wβ in (1.3) is then a Laguerre weight. More generally, all matrix ensembles with eigenvalue probability de... |

133 | Correlation functions, cluster functions, and spacing distributions for random matrices,”
- Tracy, Widom
- 1998
(Show Context)
Citation Context ...ows closely the work of the first two authors who showed in [7, 8] analogous results for Hermite-type ensembles. As in [7, 8] we use the version of the orthogonal polynomial method presented in [25], =-=[22]-=- to analyze the local eigenvalue statistics. The necessary asymptotic information on the Laguerre-type orthogonal polynomials is taken from [23]. 1. Introduction In this paper we consider ensembles of... |

82 | Nonstandard symmetry classes in mesoscopic normal–superconducting hybrid structures Phys.
- Altland, Zirnbauer
- 1997
(Show Context)
Citation Context .../2 (see e.g. [15]). In physics, Laguerre ensembles emerge e.g. in the study of Dirac operators in quantum chromodynamics and in the study of disordered superconductors in mesoscopic physics, see e.g. =-=[4, 24]-=-. Here we encounter not only Wishart ensembles but also random matrices with a 2 × 2 block structure which lead again to an eigenvalue probability density function of ( the form (1.2), (1.3). For exam... |

55 | Universality at the edge of the spectrum for unitary, orthogonal, and symplectic ensembles of random matrices.
- Deift, Gioev
- 2007
(Show Context)
Citation Context ...e the same for Hermite-type and Laguerre-type ensembles both in the bulk and at the soft edge. We start by stating our results for the hard edge, a case which is not present in Hermite-type ensembles =-=[7, 8]-=-. Notational remark. In Theorem 1.1 and also in other situations where we consider the hard edge, we will use the notation that an estimate holds uniformly for ξ, η in bounded subsets of (0, ∞). By th... |

47 |
Correlation functions of random matrix ensembles related to classical orthogonal polynomials.
- Nagao, Taro, et al.
- 1992
(Show Context)
Citation Context ...onsidered 1 . Consequently, the limiting local eigenvalue statistics agree for all ensembles (1.1) with the corresponding limiting statistics in the well studied classical cases of linear Q (see e.g. =-=[17, 10, 21, 16, 11]-=- and references therein). Ensembles (1.1) with linear Q are called Laguerre ensembles because wβ in (1.3) is then a Laguerre weight. More generally, all matrix ensembles with eigenvalue probability de... |

45 | Universality for eigenvalue correlations from the modified Jacobi unitary ensemble,
- Kuijlaars, Vanlessen
- 2002
(Show Context)
Citation Context ... 2 sgn(ξ − η) + O(n−τ ). This completes the proof of Theorem 1.1. ∫ ∞ √ η ( Jα+1(s) − 2α s Jα(s) ) ds ) □ Proof of Corollary 1.2(b). The case β = 2. This result can already be found in [23], see also =-=[12]-=-. Nevertheless we follow [8, Subsection 2.2] and present a somewhat different argument which is also useful for orthogonal and symplectic ensembles.UNIVERSALITY FOR ORTHOGONAL AND SYMPLECTIC LAGUERRE... |

45 | On the relation between orthogonal, symplectic and unitary matrix ensembles
- Widom
- 1999
(Show Context)
Citation Context ...f follows closely the work of the first two authors who showed in [7, 8] analogous results for Hermite-type ensembles. As in [7, 8] we use the version of the orthogonal polynomial method presented in =-=[25]-=-, [22] to analyze the local eigenvalue statistics. The necessary asymptotic information on the Laguerre-type orthogonal polynomials is taken from [23]. 1. Introduction In this paper we consider ensemb... |

34 | Spectrum of the QCD Dirac operator and chiral random matrix theory - Verbaarschot - 1994 |

31 | Universality in Random Matrix Theory for orthogonal and symplectic ensembles.
- Deift, Gioev
- 2004
(Show Context)
Citation Context ...rying weight case at the hard spectral edge was analyzed in [13] for β = 2: In this paper we do not consider varying weights. Our proof follows closely the work of the first two authors who showed in =-=[7, 8]-=- analogous results for Hermite-type ensembles. As in [7, 8] we use the version of the orthogonal polynomial method presented in [25], [22] to analyze the local eigenvalue statistics. The necessary asy... |

29 | Strong asymptotics of Laguerre-type orthogonal polynomials and applications in random matrix theory
- Vanlessen
(Show Context)
Citation Context ...) cℓ := 22−2m Am ( ) 2m − 2 , and Am := m − 1 − ℓ Further, since A12 = At 21 and Y = Y t , (2.20) yields ( n (2.23) A12 = A21 + O n βn −1/m ) . m∏ j=1 2j − 1 . 2j Proof. The proof uses the results in =-=[23]-=- on the asymptotics of the recurrence coefficients bn−1 and an appearing in the three-term recurrence relation (2.24) xφn(x) = bnφn+1(x) + anφn(x) + bn−1φn−1(x), satisfied by the orthonormal functions... |

25 |
Painlevé transcendent evaluation of the scaled distribution of the smallest eigenvalue in the Laguerre orthogonal and symplectic ensembles. arXiv:nlin/0005064
- Forrester
- 2000
(Show Context)
Citation Context ...onsidered 1 . Consequently, the limiting local eigenvalue statistics agree for all ensembles (1.1) with the corresponding limiting statistics in the well studied classical cases of linear Q (see e.g. =-=[17, 10, 21, 16, 11]-=- and references therein). Ensembles (1.1) with linear Q are called Laguerre ensembles because wβ in (1.3) is then a Laguerre weight. More generally, all matrix ensembles with eigenvalue probability de... |

25 | Universality for eigenvalue correlations at the origin of the spectrum,
- Kuijlaars, Vanlessen
- 2003
(Show Context)
Citation Context ...mallest eigenvalues. Corresponding results for unitary (β = 2) Laguerre-type ensembles have been proved by the fourth author in [23]. The varying weight case at the hard spectral edge was analyzed in =-=[13]-=- for β = 2: In this paper we do not consider varying weights. Our proof follows closely the work of the first two authors who showed in [7, 8] analogous results for Hermite-type ensembles. As in [7, 8... |

23 |
Level-spacing distributions and the Bessel kernel
- Tracy, Widom
- 1994
(Show Context)
Citation Context |

21 |
Asymptotic correlations at the spectrum edge of random matrices
- Nagao, Forrester
- 1995
(Show Context)
Citation Context |

16 | versus Pfaff lattice and related polynomials - Adler, Moerbeke, et al. - 2002 |

11 | Universality of random matrices in the microscopic limit and the Dirac operator spectrum
- Akemann, Damgaard, et al.
- 1997
(Show Context)
Citation Context ...n [23] play a key role in our proof of universality for β = 1, 4. Universality for Laguerre-type ensembles, for all three cases β = 1, 2 and 4, has been considered in the physics literature (see e.g. =-=[3, 18]-=- and references therein). More information on the history of universality for matrix ensembles can be found the introductions of [7, 8] and in [6]. The basic structure of the proof in this paper is si... |

9 |
M.: Universality in chiral random matrix theory at β = 1 and β
- Sener, Verbaarschot
- 1998
(Show Context)
Citation Context ...n [23] play a key role in our proof of universality for β = 1, 4. Universality for Laguerre-type ensembles, for all three cases β = 1, 2 and 4, has been considered in the physics literature (see e.g. =-=[3, 18]-=- and references therein). More information on the history of universality for matrix ensembles can be found the introductions of [7, 8] and in [6]. The basic structure of the proof in this paper is si... |

6 |
Universality for orthogonal and symplectic ensembles of random matrices, with generalized Laguerre type weights, in preparation
- Deift, Gioev, et al.
(Show Context)
Citation Context ...is to show the invertibility of a certain m ×m matrix (see Tm in (2.49) below), where m denotes the degree of the polynomial Q. This will be done in Section 3. Here estimates (essentially) derived in =-=[7, 5]-=- are very useful (see Propositions 3.4, 3.5, 3.6). However, the proof of the invertibility of the m × m matrix Tm in the present situation, isUNIVERSALITY FOR ORTHOGONAL AND SYMPLECTIC LAGUERRE-TYPE ... |

2 |
Universality for mathematical and physical systems,” math-ph/0603038
- Deift
(Show Context)
Citation Context ...en considered in the physics literature (see e.g. [3, 18] and references therein). More information on the history of universality for matrix ensembles can be found the introductions of [7, 8] and in =-=[6]-=-. The basic structure of the proof in this paper is similar to [7, 8] and relies on the orthogonal polynomial method developed in [22] and [25]. A detailed description of the strategy of proof can be ... |