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## Products and ratios of characteristic polynomials of random Hermitian matrices (2003)

Venue: | J. Math. Phys |

Citations: | 35 - 5 self |

### Citations

933 |
Random Matrices
- Mehta
- 1967
(Show Context)
Citation Context ...cteristic polynomials of random Hermitian matrices that have appeared recently in the literature. 1 1 Introduction In random matrix theory, unitary ensembles of N N matrices fHg play a central role [16]. Such ensembles are described by a measure d withsnite moments R R jxj k d(x)s1, k = 0; 1; 2; , and the distribution function for the eigenvalues fx i = x i (H)g of matrices H in the ensemble... |

527 |
Orthogonal polynomials and random matrices: a Riemann–Hilbert approach,
- Deift
- 1999
(Show Context)
Citation Context ...here b 0 0. Conversely, modulo certain essential self-adjointness issues, d is the spectral measure for J in the cyclic subspace generated by J and the vector e 1 = (1; 0; 0; ) T (see, e.g., [4]). It follows that the transformation of measures d ! d [`;m] (1.10) leads to the transformation of operators J(d) ! J(d [`;m] ): (1.11) For appropriate choices of 1 ; ; m and 1 ; ... |

342 |
Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory,
- Deift, Kriecherbauer, et al.
- 1999
(Show Context)
Citation Context ... asymptotically using the non-commutative steepest-descent method introduced by Deift and Zhou [8], and further developed with Venakides in [7] to allow for fully non-linear oscillations, and in [6], =-=[5]-=-. Our goal in this paper is to give new, streamlined proofs of (2.6)-(3.12), using only the properties of orthogonal polynomials and a minimum of combinatorics. Along the way we 2 will also need an in... |

303 | A steepest descent method for oscillatory Riemann–Hilbert problems. Asymptotics for the MKdV equation,
- Deift, Zhou
- 1993
(Show Context)
Citation Context ...m as a Riemann-Hilbert problem by Fokas, Its and Kitaev [9]. The Riemann-Hilbert problem is then analyzed asymptotically using the non-commutative steepest-descent method introduced by Deift and Zhou =-=[8]-=-, and further developed with Venakides in [7] to allow for fully non-linear oscillations, and in [6], [5]. Our goal in this paper is to give new, streamlined proofs of (2.6)-(3.12), using only the pro... |

236 |
Strong asymptotics of orthogonal polynomials with respect to exponential weights,
- Deift, Kriecherbauer, et al.
- 1999
(Show Context)
Citation Context ...lyzed asymptotically using the non-commutative steepest-descent method introduced by Deift and Zhou [8], and further developed with Venakides in [7] to allow for fully non-linear oscillations, and in =-=[6]-=-, [5]. Our goal in this paper is to give new, streamlined proofs of (2.6)-(3.12), using only the properties of orthogonal polynomials and a minimum of combinatorics. Along the way we 2 will also need ... |

96 |
New result in small dispersion KdV by an extension of the steepest descent method for Riemann-Hilbert problems
- Deift, Venakides, et al.
- 1985
(Show Context)
Citation Context ...and Kitaev [9]. The Riemann-Hilbert problem is then analyzed asymptotically using the non-commutative steepest-descent method introduced by Deift and Zhou [8], and further developed with Venakides in =-=[7]-=- to allow for fully non-linear oscillations, and in [6], [5]. 2Our goal in this paper is to give new, streamlined proofs of (2.6)-(3.12), using only the properties of orthogonal polynomials and a min... |

75 |
Discrete Painleve equations and their appearance in quantum gravity.
- Fokas, Its, et al.
- 1991
(Show Context)
Citation Context ...cial cases, we again refer the reader to [11]. The asymptotic analysis in [11] is based on the reformulation of the orthogonal polynomial problem as a Riemann-Hilbert problem by Fokas, Its and Kitaev =-=[9]-=-. The Riemann-Hilbert problem is then analyzed asymptotically using the non-commutative steepest-descent method introduced by Deift and Zhou [8], and further developed with Venakides in [7] to allow f... |

70 | Characteristic polynomials of random matrices
- Brezin, Hikami
- 2000
(Show Context)
Citation Context ...ndom matrices with respect to various ensembles. Such averages are used, in particular, in making predictions about the moments of the Riemann-zeta function, see [15, 14, 13] (circular ensembles) and [3] (unitary ensembles). Many other uses are described, for example, in [1], [12] and [11]. By (1.2), for unitary ensembles, such averages have the form Q K j=1 DN [ j ; H] Q M j=1 DN [ j ; H] = 1... |

60 | On the characteristic polynomials of a random unitary matrix
- Hughes, Keating, et al.
- 2001
(Show Context)
Citation Context ... DN [; H] = N Y i=1 ( x i (H)) of random matrices with respect to various ensembles. Such averages are used, in particular, in making predictions about the moments of the Riemann-zeta function, see [1=-=5, 14, 1-=-3] (circular ensembles) and [3] (unitary ensembles). Many other uses are described, for example, in [1], [12] and [11]. By (1.2), for unitary ensembles, such averages have the form Q K j=1 DN [ j ; H]... |

36 | E.: An exact formula for general spectral correlation functions of random matrices - Fyodorov, Strahov - 2003 |

34 | Universal Results for Correlations of Characteristic Polynomials
- Strahov
(Show Context)
Citation Context ..., in making predictions about the moments of the Riemann-zeta function, see [15, 14, 13] (circular ensembles) and [3] (unitary ensembles). Many other uses are described, for example, in [1], [12] and [11]. By (1.2), for unitary ensembles, such averages have the form Q K j=1 DN [ j ; H] Q M j=1 DN [ j ; H] = 1 ZN Z Z Q K j=1 Q N i=1 ( j x i ) Q M j=1 Q N i=1 ( j x i ) (x) 2 d(x): (1.3)... |

28 |
Orthogonal Polynomials, volume 23
- Szego
- 1975
(Show Context)
Citation Context ... k (x)d(x) = c j c ksjk ; j; k 0; (1.4) where the norming constants c j 's are positive. The key observation in our approach is that for K = 1 and M = 0 in (1.3) DN [; H] = N () (1.5) (see [18]). In our words, the orthogonal polynomial N () with respect to d is also precisely the average polynomial N Y i=1 ( x i ) with respect to d P ;N . Formula (1.5) appears already in the work of He... |

24 | Developments in random matrix theory,
- Forrester, Snaith, et al.
- 2003
(Show Context)
Citation Context ... DN [; H] = N Y i=1 ( x i (H)) of random matrices with respect to various ensembles. Such averages are used, in particular, in making predictions about the moments of the Riemann-zeta function, see [1=-=5, 14, 1-=-3] (circular ensembles) and [3] (unitary ensembles). Many other uses are described, for example, in [1], [12] and [11]. By (1.2), for unitary ensembles, such averages have the form Q K j=1 DN [ j ; H]... |

23 |
Normand: Moments of the characteristic polynomial in the three ensembles of random matrices
- Mehta
- 2001
(Show Context)
Citation Context ...x) 2 d(x): (1.3) In this paper we consider certain explicit determinantal formulae for (1.3) { see (2.6), (2.24), (2.36), (3.3), (3.12) below. Formula (2.6) is due to Brezin and Hikami [3] (see also [=-=17-=-], and when all the j 's are equal, see [10]), whereas (2.24), (2.36), (3.3) and (3.12) are due to Fyodorov and Strahov [12, 11]. The papers [12, 11] also contain a discussion of the history of these... |

20 |
Random matrix theory and ζ(1
- Keating, Snaith
(Show Context)
Citation Context ...mials DN[µ, H] = (µ − xi(H)) of random matrices with respect i=1 to various ensembles. Such averages are used, in particular, in making predictions about the moments of the Riemann-zeta function, see =-=[15, 14, 13]-=- (circular ensembles) and [3] (unitary ensembles). Many other uses are described, for example, in [1], [12] and [11]. By (1.2), for unitary ensembles, such averages have the form 〈∏ K j=1 DN[µj, H] ∏ ... |

16 |
Note sur une relation les integrales définies des produits de fonctions, Mém. de la Soc
- Andréief
(Show Context)
Citation Context ...his relation is not immediately clear, and requires further algebraic manipulation. 3 Formulae of two-point function type The following integral version of the Binet-Cauchy formula is due to Andréief =-=[2]-=-, and plays a basic role in our calculations. Lemma 3.1. Let (X, dµ) be a measure space and suppose fi, gj ∈ L2 (X, dµ) for 1 ≤ i, j ≤ k. Then ∫ ∫ · · · det(fi(xj))1≤i,j≤k det(gi(xj))1≤i,j≤kdµ(x1) · ·... |

15 |
The connection between systems of polynomials orthogonal with respect to different distribution functions.
- UvARov
- 1969
(Show Context)
Citation Context ...e above considerations for the case d [`;m] (t) = ( 1 t) ( ` t) ( 1 t) ( m t) d(t): (2.31) Thesrst result is a Christoel type formula for the measure (2.31), which is due to Uvarov [19]: Lemma 2.11. Suppose 0 m n. Then the monic orthogonal polynomials [`;m] n (t)'s with respect to the measure d `;m] (t) have the following representation: [`;m] n (t) = 1 (t ` ) : : : (t ... |

12 |
Results in Small Dispersion KdV by an Extension of the Steepest Descent Method for Riemann-Hilbert Problems
- Deift, Zhou
- 1997
(Show Context)
Citation Context ...and Kitaev [9]. The Riemann-Hilbert problem is then analyzed asymptotically using the non-commutative steepest-descent method introduced by Deift and Zhou [8], and further developed with Venakides in =-=[7]-=- to allow for fully non-linear oscillations, and in [6], [5]. Our goal in this paper is to give new, streamlined proofs of (2.6)-(3.12), using only the properties of orthogonal polynomials and a minim... |

9 | Correlators of spectral determinants in quantum chaos
- Andreev, Simons
- 1995
(Show Context)
Citation Context ... in particular, in making predictions about the moments of the Riemann-zeta function, see [15, 14, 13] (circular ensembles) and [3] (unitary ensembles). Many other uses are described, for example, in [1], [12] and [11]. By (1.2), for unitary ensembles, such averages have the form Q K j=1 DN [ j ; H] Q M j=1 DN [ j ; H] = 1 ZN Z Z Q K j=1 Q N i=1 ( j x i ) Q M j=1 Q N i=1 ( j x i ) (x)... |

3 |
Application of the tau-function theory of Painlevé equations to random matrices
- Forrester, Witte
(Show Context)
Citation Context ...ertain explicit determinantal formulae for (1.3) { see (2.6), (2.24), (2.36), (3.3), (3.12) below. Formula (2.6) is due to Brezin and Hikami [3] (see also [17], and when all the j 's are equal, see [=-=10]-=-), whereas (2.24), (2.36), (3.3) and (3.12) are due to Fyodorov and Strahov [12, 11]. The papers [12, 11] also contain a discussion of the history of these formulae. The formulae (3.3) and (3.12) are ... |

2 |
Orthogonal Polynomials, volume 23 of American Mathematical Society, Colloquium Publications
- Szegö
- 1975
(Show Context)
Citation Context ... πj(x)πk(x)dα(x) = cjckδjk, j, k ≥ 0, (1.4) R where the norming constants cj’s are positive. The key observation in our approach is that for K = 1 and M = 0 in (1.3) 〈 DN[µ, H] 〉 α = πN(µ) (1.5) (see =-=[18]-=-). In our words, the orthogonal polynomial πN(µ) with respect to dα is also precisely N∏ the average polynomial (µ − xi) with respect to d Pα,N. Formula (1.5) appears already in i=1 the work of Heine ... |

1 |
Note sur une relation les integrales de des produits des fonctions. Mem. de la Soc. Sci
- Andreief
(Show Context)
Citation Context ...is relation is not immediately clear, and requires further algebraic manipulation. 3 Formulae of two-point function type The following integral version of the Binet-Cauchy formula is due to Andreief [2], and plays a basic role in our calculations. Lemma 3.1. Let (X; d) be a measure space and suppose f i ; g j 2 L 2 (X; d) for 1 i; j k. Then Z X Z X det(f i (x j )) 1i;jk det(g i (x j ))... |

1 |
Random matrix thoery and the derivative of the Riemann-zeta function
- Hughes, Keating, et al.
- 2000
(Show Context)
Citation Context ... DN [; H] = N Y i=1 ( x i (H)) of random matrices with respect to various ensembles. Such averages are used, in particular, in making predictions about the moments of the Riemann-zeta function, see [1=-=5, 14, 1-=-3] (circular ensembles) and [3] (unitary ensembles). Many other uses are described, for example, in [1], [12] and [11]. By (1.2), for unitary ensembles, such averages have the form Q K j=1 DN [ j ; H]... |