Perturbed Hankel determinant, correlation functions and Painlevé equations

Chen, Min; Chen, Yang; Fan, Engui

doi:10.1063/1.4939276

Cited by 13 publications

(26 citation statements)

References 46 publications

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“…Remark As

n \to \infty

and

t = 0

, the asymptotic behavior of the recurrence coefficients in the above and are coherent with the classical theory in, see also and appendix C therein. The above third‐order differential equation is integrable, which had been shown in the paper, and

y false(ς false) = ς r^{'} (ς)

satisfies a Painlevé III equation.…”

Section: The Proofs Of Theoremssupporting

confidence: 81%

“…Remark The singular factor

e^{- t / x}, t > 0

, introduces an infinitely strong zero at 0 and has the effect of pushing the left end point of the equilibrium density away from 0 at a slow speed. If

bolda

is left end point of the support of the density, then for

t > 0

, and large n ,

a = \frac{t^{\frac{2}{3}}}{{false(2 (2 n + α + β) false)}^{\frac{2}{3}}} false(1 + O (n^{- 2 / 3}) false) = \frac{ς^{\frac{2}{3}}}{4 n^{2}} (1 + O false(n^{- 2} false)), ς = 2 n^{2} t,

which follows from a combination of (2.25) and (2.26) in . For the shifted Jacobi weight

w false(x false) = x^{α} {(1 - x)}^{β}, a = \frac{α^{2 / 3}}{{false(2 n + α + β false)}^{2}} false(1 + O (n^{- 2}) false)

.…”

Section: Deift–zhou Steepest Descent Analysismentioning

confidence: 97%

“…With the definition of

H_{n} false(t false)

in and integrating once again, then one derives the formula . Moreover, as

t = 0

, the Pollaczek–Jacobi weight reduces to the shifted Jacobi weight function and its Hankel determinant calculated by

D_{n} false(w, 0 false) = \prod_{j = 0}^{n - 1} h_{j} false(0 false)

, where

h_{n} false(0 false)

is given by

h_{n} false(0 false) = \frac{n! Γ (n + 1 + α) Γ (n + 1 + β) Γ (n + 1 + α + β)}{Γ (2 n + 1 + α + β) Γ (2 n + 2 + α + β)},

it can be directly derived from ( C 13) in . Inserting the facts of

H_{n} false(t false) = H false(ς false) false(1 + O (n^{- \frac{2}{3}}) false), H_{n}^{'} false(t false) = 2 n^{2} {scriptH}^{'} false(ς false) false(1 + O (n^{- \frac{2}{3}}) false)

, and

H_{n}^{''} false(t false) = 4 n^{4} {scriptH}^{''} false(ς false) false(1 + O (n^{- \frac{2}{3}}) false)

into (32), collecting the coefficients of 4 n 4 , then one obtains the σ‐form Painlevé equation (35).…”

Section: The Proofs Of Theoremsmentioning

confidence: 99%

“…which follows from a combination of (2.25) and (2.26) in. 29 For the shifted Jacobi weight ( ) = (1 − ) , = 2∕3 (2 + + ) 2 (1 + ( −2 )). For → ∞, one recalls the conformal mapping = 2 0 ( ) = −4 2 (1 + ( )) in (85), rescaling = 2 3 in (90), one finds = − 2 3 as = −1.…”

Section: (C)mentioning

confidence: 99%

“…With these asymptotic formulas, it is easy to verify the uniform asymptotic formula (39) by (28) for ( ). Similarly, the uniform asymptotic expansion for ( ) follows from (29) and (143), and inserting it into the Painlevé V equation (30), then the third-order integrable equation (42) is derived from collecting the coefficients of −2 . ■ Remark 5.…”

Section: The Proofs Of Theoremsmentioning

confidence: 99%

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The Riemann–Hilbert analysis to the Pollaczek–Jacobi type orthogonal polynomials

Chen

Fan

2019

Stud Appl Math

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In this paper, we study polynomials orthogonal with respect to a Pollaczek–Jacobi type weight truerightleftwpJ(x,t)=e−txxαfalse(1−xfalse)β,1emt≥0,rightleft1emα>0,1emβ>0,1emx∈false[0,1false].The uniform asymptotic expansions for the monic orthogonal polynomials on the interval (0,1) and outside this interval are obtained. Moreover, near x=0, the uniform asymptotic expansion involves Airy function as ς=2n2t→∞,n→∞, and Bessel function of order α as ς=2n2t→0,n→∞; in the neighborhood of x=1, the uniform asymptotic expansion is associated with Bessel function of order β as n→∞. The recurrence coefficients and leading coefficient of the orthogonal polynomials are expressed in terms of a particular Painlevé III transcendent. We also obtain the limit of the kernel in the bulk of the spectrum. The double scaled logarithmic derivative of the Hankel determinant satisfies a σ‐form Painlevé III equation. The asymptotic analysis is based on the Deift and Zhou's steepest descent method.

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“…Remark As

n \to \infty

and

t = 0

y false(ς false) = ς r^{'} (ς)

satisfies a Painlevé III equation.…”

Section: The Proofs Of Theoremssupporting

confidence: 81%

“…Remark The singular factor

e^{- t / x}, t > 0

, introduces an infinitely strong zero at 0 and has the effect of pushing the left end point of the equilibrium density away from 0 at a slow speed. If

bolda

is left end point of the support of the density, then for

t > 0

, and large n ,

a = \frac{t^{\frac{2}{3}}}{{false(2 (2 n + α + β) false)}^{\frac{2}{3}}} false(1 + O (n^{- 2 / 3}) false) = \frac{ς^{\frac{2}{3}}}{4 n^{2}} (1 + O false(n^{- 2} false)), ς = 2 n^{2} t,

which follows from a combination of (2.25) and (2.26) in . For the shifted Jacobi weight

w false(x false) = x^{α} {(1 - x)}^{β}, a = \frac{α^{2 / 3}}{{false(2 n + α + β false)}^{2}} false(1 + O (n^{- 2}) false)

.…”

Section: Deift–zhou Steepest Descent Analysismentioning

confidence: 97%

“…With the definition of

H_{n} false(t false)

in and integrating once again, then one derives the formula . Moreover, as

t = 0

, the Pollaczek–Jacobi weight reduces to the shifted Jacobi weight function and its Hankel determinant calculated by

D_{n} false(w, 0 false) = \prod_{j = 0}^{n - 1} h_{j} false(0 false)

, where

h_{n} false(0 false)

is given by

h_{n} false(0 false) = \frac{n! Γ (n + 1 + α) Γ (n + 1 + β) Γ (n + 1 + α + β)}{Γ (2 n + 1 + α + β) Γ (2 n + 2 + α + β)},

it can be directly derived from ( C 13) in . Inserting the facts of

H_{n} false(t false) = H false(ς false) false(1 + O (n^{- \frac{2}{3}}) false), H_{n}^{'} false(t false) = 2 n^{2} {scriptH}^{'} false(ς false) false(1 + O (n^{- \frac{2}{3}}) false)

, and

H_{n}^{''} false(t false) = 4 n^{4} {scriptH}^{''} false(ς false) false(1 + O (n^{- \frac{2}{3}}) false)

into (32), collecting the coefficients of 4 n 4 , then one obtains the σ‐form Painlevé equation (35).…”

Section: The Proofs Of Theoremsmentioning

confidence: 99%

Section: (C)mentioning

confidence: 99%

Section: The Proofs Of Theoremsmentioning

confidence: 99%

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The Riemann–Hilbert analysis to the Pollaczek–Jacobi type orthogonal polynomials

Chen

Fan

2019

Stud Appl Math

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Asymptotic gap probability distributions of the Gaussian unitary ensembles and Jacobi unitary ensembles

2018

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In this paper, we address a class of problems in unitary ensembles. Specifically, we study the probability that a gap symmetric about 0, i.e. (−a, a) is found in the Gaussian unitary ensembles (GUE) and the Jacobi unitary ensembles (JUE) (where in the JUE, we take the parameters α = β). By exploiting the even parity of the weight, a doubling of the interval to (a 2 , ∞) for the GUE, and (a 2 , 1), for the (symmetric) JUE, shows that the gap probabilities maybe determined as the product of the smallest eigenvalue distributions of the LUE with parameter α = −1/2, and α = 1/2 and the (shifted) JUE with weights x 1/2 (1 − x) β andThe σ function, namely, the derivative of the log of the smallest eigenvalue distributions of the finite-n LUE or the JUE, satisfies the Jimbo-Miwa-Okamoto σ form of P V and P V I , although in the shift Jacobi case, with the weight x α (1 − x) β , the β parameter does not show up in the equation. We also obtain the asymptotic expansions for the smallest eigenvalue distributions * lvshulin1989@163.com † yangbrookchen@yahoo.co.uk ‡ Corresponding author and faneg@fudan.edu.cn of the Laguerre unitary and Jacobi unitary ensembles after appropriate double scalings, and obtained the constants in the asymptotic expansion of the gap probablities, expressed in term of the Barnes G− function valuated at special point. IntroductionIn the work of Adler and Van Moerbeke [1], the largest eigenvalue distribution of ensembles of n × n random matrices generated by Gaussian, Laguerre and Jacobi weights for general values of the symmetry parameter β, (not to be confused with the β parameter in the Jacobi weight), has been systematically studied, from the perspective of differential operators involving multiple time variables.The gap probabilities that are studied in this paper, the unitary case, denoted by P(a, n), are represented as Fredholm determinant of an integral operator, from the early papers of Mehta, Gaudin and Dyson [19], [20], [22], [31], and [32]. In [28], the gap probability, where a union of disjoint intervals is free of eigenvalues, the integral operator has the sine kernel K(x, y) := sin(x−y) π(x−y) . The (multiple-) gap probability itself was obtained in an expansion in terms of the resolvent of the integral equation. In a tour de force computations, JMMS showed that in the single interval case where (−b, b) is free of eigenvalues, the quantity σ(τ ) := τ d dτ logdet(I − K (−b,b) ), τ = 2b, satisfies a second order non-linear differential equation. Tracy and Widom [36] studied, the finite n version of such problems, namely, the distribution of the smallest eigenvalue in the Laguerre unitary ensembles, and the largest eigenvalue distribution of the Gaussian unitary ensembles starting from the Christoffel-Darboux or Reproducing Kernel, constructed out of the "natural" orthogonal polynomials, namely the Laguerre and Hermite polynomials, respectively. Through a series of differentiation formulas, Tracy and Widom found the finite n version of the differential equations satisfied by the resolvent a...

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Painlevé III′ and the Hankel determinant generated by a singularly perturbed Gaussian weight

2018

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In this paper, we study the Hankel determinant generated by a singularly perturbed Gaus-By using the ladder operator approach associated with the orthogonal polynomials, we show that the logarithmic derivative of the Hankel determinant satisfies both a non-linear second order difference equation and a non-linear second order differential equation. The Hankel determinant also admits an integral representation involving a Painlevé III ′ . Furthermore, we consider the asymptotics of the Hankel determinant under a double scaling, i.e. n → ∞ and t → 0 such that s = (2n + 1)t is fixed. The asymptotic expansions of the scaled Hankel determinant for large s and small s are established, from which Dyson's constant appears.

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Perturbed Hankel determinant, correlation functions and Painlevé equations

Cited by 13 publications

References 46 publications

The Riemann–Hilbert analysis to the Pollaczek–Jacobi type orthogonal polynomials

The Riemann–Hilbert analysis to the Pollaczek–Jacobi type orthogonal polynomials

Asymptotic gap probability distributions of the Gaussian unitary ensembles and Jacobi unitary ensembles

Painlevé III′ and the Hankel determinant generated by a singularly perturbed Gaussian weight

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