Gap Probabilities for Double Intervals in Hermitian Random Matrix Ensembles as  -Functions – Spectrum Singularity Case

Witte, N. S.

doi:10.1023/b:math.0000045556.53148.02

Cited by 6 publications

(6 citation statements)

References 22 publications

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“…Similar τ -function identities to (5.91) have occurred in the works [25], [70]. The Painlevé transcendent evaluations of E LOE 1 ((s, ∞); N ; 0) and E LSE 4 ((s, ∞); N/2; 0) given in [5] differ from (5.86) and (5.87), involving instead of τ -functions for the PIII' system, the transformed PV transcendent found in [65] to be simply related to the derivative of…”

supporting

confidence: 69%

“…It is interesting to compare (5.88) with the structural formula[27] (s, ∞);N ; 0) + E LUE (s, ∞); N ; 0) E LOE (s, ∞); N ; 0) . (5.89)Doing this tells us thatE LUE (s, ∞); N ; 0) = τ III ′ ((s/4) 2 ) v 1 =N +1/III ′ ((s/4) 2 ) v 1 =N −1/t) satisfies (4.8), and soτV (s) ν 0 =ν 1 =0 ν 2 =ν 3 =N = τ III ′ ((s/4) 2 ) v 1 =N +1/III ′ ((s/4) 2 ) v 1 =N −1/(5.92)Similar τ -function identities to (5.92) have occured in the works[25,70]. The Painlevé transcendent evaluations of E (s, ∞); N ; 0) and E LSE 4…”

mentioning

confidence: 59%

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Application of theτ-function theory of Painlevé equations to random matrices:P_VI, the JUE, CyUE, cJUE and scaled limits

Forrester¹,

Witte²

2004

Nagoya Mathematical Journal

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Okamoto has obtained a sequence of τ-functions 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 re-expressed 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 non-negative integer) and Laguerre symplectic ensemble (LSE) (parameter a an even non-negative integer) as finite dimensional combinatorial integrals over the symplectic and orthogonal groups respectively; to the evaluation of the cumulative distribution function for the last passage time in certain models of directed percolation; to the τ-function evaluation of the largest eigenvalue in the finite LOE and LSE with parameter a = 0; and to the characterisation of the diagonal-diagonal spin-spin correlation in the two-dimensional Ising model.

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supporting

confidence: 69%

mentioning

confidence: 59%

Application of theτ-function theory of Painlevé equations to random matrices:P_VI, the JUE, CyUE, cJUE and scaled limits

Forrester¹,

Witte²

2004

Nagoya Mathematical Journal

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195

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“…x=4N [8] where u satisfies the s form of the Painlevé III 0 di¤erential equation In the recent work [26] this identity between transcendents is interpreted as an identity between the Hamiltonian for a Painlevé III system, and the sum of Hamiltonians for particular Painlevé III 0 systems. With the weight (2.38), and p ¼ 1, t 0 :¼ 0, t 1 :¼ t so that I ft 2i ; t 2iþ1 g ¼ ðÀt; tÞ; I þ ft 2i ; t 2iþ1 g ¼ ð0; tÞ; Hence in the cases a À 1=2 A Z b0 the right hand side of (2.39) is equal to the product of an ða À 1=2Þ Â ða À 1=2Þ and ða þ 1=2Þ Â ða þ 1=2Þ determinant.…”

Section: Generalized Hermite Ensemblementioning

confidence: 99%

Evenness symmetry and inter-relationships between gap probabilities in random matrix theory

Forrester

2006

Forum Mathematicum

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Our interest is in the generating function E N; b ðI ; x; w b Þ for the probabilities E N; b ðn; I ; w b Þ that in a matrix ensemble with unitary ðb ¼ 2Þ or orthogonal ðb ¼ 1) symmetry, characterized by the weight w b ðlÞ and having N eigenvalues, the interval I contains exactly n eigenvalues. Using a determinant formula for E N; 2 , a general quadratic identity is obtained which relates E N; 2 in the case I and w 2 ðxÞ even to a product of generating functions E N; 2 with di¤erent I , w 2 ðlÞ and N, and for which the eigenvalues are positive. Also, generalizing some earlier calculations, the sum E N; 1 ð2n À 1; I; w 1 Þ þ E N; 1 ð2n; I ; w 1 Þ for N even, I ¼ ðÀt; tÞ and w 1 an even classical weight is shown to equal E N=2; 2 ðn; ð0; t 2 Þ; w 2 Þ for w 2 related to w 1 . Implications of these identities are discussed. 2000 Mathematics Subject Classification: 15A52; 34M55. P. J. Forrester 712 Brought to you by | University of Arizona Authenticated Download Date | 6/1/15 3:53 AM 1 In [10] the minus sign before has been misprinted as a plus; I thank A. Its for pointing this out.Gap probabilities in random matrix theory 713Brought to you by |

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“…Actually, an analogue of this class of transformations was noted by Okamoto [20] in the case of the fourth transcendent, but appeared not to be appreciated as such. Aside from the intrinsic interest of these transformations there are some practical motivations from the theory of random matrices where they arise as multiplicative identities for the probabilities that certain spectral intervals are free of eigenvalues [24].…”

Section: N S Witte (Communicated By Mark J Ablowitz)mentioning

confidence: 99%

New transformations for Painlevé’s third transcendent

Witte

2004

Proc. Amer. Math. Soc.

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Abstract. We present transformations relating the third transcendent of Painlevé with parameter sets located at the corners of the Weyl chamber for the symmetry group of the system, the affine Weyl group of the root system B (1) 2 , to those at the origin. This transformation entails a scaling of the independent variable, and implies additive identities for the canonical Hamiltonians and product identities for the τ -functions with these parameter sets.A curious anomaly has existed in the theory of the Painlevé transcendents for many years, for almost a century in fact. This anomaly is the existence of a particular transformation for the second Painlevé transcendent q(t; α + 1 / 2 ) that had no known analogue for any of the other transcendents. One recalls that the second Painlevé transcendent q = q(t; α + 1 / 2 ) is by definition a solution of the P II differential equation 1 , and usingtransformations of this group, solutions for arbitrary v 1 can be related to a solution whose parameter v 1 ∈ (0, 1 / 2 ] (or in [ 1 / 2 , 1)). This interval defines the Weyl chamber and the above transformation in question relates the solutions at the endpoints of this interval (see Figure 1).Specifically the transformation takes the form

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Gap Probabilities for Double Intervals in Hermitian Random Matrix Ensembles as -Functions – Spectrum Singularity Case

Cited by 6 publications

References 22 publications

Application of theτ-function theory of Painlevé equations to random matrices:P_VI, the JUE, CyUE, cJUE and scaled limits

Application of theτ-function theory of Painlevé equations to random matrices:P_VI, the JUE, CyUE, cJUE and scaled limits

Evenness symmetry and inter-relationships between gap probabilities in random matrix theory

New transformations for Painlevé’s third transcendent

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Gap Probabilities for Double Intervals in Hermitian Random Matrix Ensembles as -Functions – Spectrum Singularity Case

Cited by 6 publications

References 22 publications

Application of theτ-function theory of Painlevé equations to random matrices:PVI, the JUE, CyUE, cJUE and scaled limits

Application of theτ-function theory of Painlevé equations to random matrices:PVI, the JUE, CyUE, cJUE and scaled limits

Evenness symmetry and inter-relationships between gap probabilities in random matrix theory

New transformations for Painlevé’s third transcendent

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Product

Resources

About

Application of theτ-function theory of Painlevé equations to random matrices:P_VI, the JUE, CyUE, cJUE and scaled limits

Application of theτ-function theory of Painlevé equations to random matrices:P_VI, the JUE, CyUE, cJUE and scaled limits