Singular linear statistics of the Laguerre unitary ensemble and Painlevé. III. Double scaling analysis

Chen, Min; Chen, Yang

doi:10.1063/1.4922620

Cited by 26 publications

(25 citation statements)

References 40 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Remark Let

\hat{scriptH} false(ς false) = H false(2 ς false) - α^{2} / 4

, then

\hat{scriptH} false(ς false)

satisfies the Painlevé III σ‐form equation obtained by Ohyama–Kawamuko–Sakai–Okamoto, see (18) in, see also the Remark in . We note that and are the same as the double scaling limit of the Hankel determinant associated with the singularly perturbed Laguerre weight, see theorem in, the σ‐form Painlevé equation is equivalent to the second‐order eq.…”

Section: The Proofs Of Theoremsmentioning

confidence: 85%

“…Let( ) = (2 ) − 2 ∕4, then( ) satisfies the Painlevé III -form equation obtained by Ohyama-Kawamuko-Sakai-Okamoto, see (18) in, 30 see also the Remark 3 in. 31 We note that (34) and (36) are the same as the double scaling limit of the Hankel determinant associated with the singularly perturbed Laguerre weight, see theorem 1 in, 13 the -form Painlevé equation (35) is equivalent to the second-order eq. (30) in, 13 but the double scaling scheme and initial data ( , 0) are different from each other.…”

mentioning

confidence: 99%

See 1 more Smart Citation

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

Chen

Fan

2019

Stud Appl Math

Self Cite

View full text Add to dashboard Cite

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.

show abstract

“…Remark Let

\hat{scriptH} false(ς false) = H false(2 ς false) - α^{2} / 4

, then

\hat{scriptH} false(ς false)

Section: The Proofs Of Theoremsmentioning

confidence: 85%

mentioning

confidence: 99%

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

Chen

Fan

2019

Stud Appl Math

Self Cite

View full text Add to dashboard Cite

show abstract

“…See [30]. The C potential (2.22) introduced in [9], after a change of variable, satisfies the above equation.…”

Section: An Equivalent Expression Readsmentioning

confidence: 99%

Perturbed Hankel determinant, correlation functions and Painlevé equations

Chen¹,

Chen²,

Fan

2016

Journal of Mathematical Physics

Self Cite

View full text Add to dashboard Cite

We continue with the study of the Hankel determinant, D n (t, α, β) := det , generated by a Pollaczek-Jacobi type weight,This reduces to the "pure" Jacobi weight at t = 0. We may take α ∈ R , in the situation while t is strictly greater than 0. It was shown in Chen and Dai (2010), that the logarithmic derivative of this Hankel determinant satisfies a Jimbo-Miwa-Okamoto σ -form of Painlevé V ( P V ). In fact the logarithmic of the Hankel determinant has an integral representation in terms of a particular P V . In this paper, we show that, under a double scaling, where n the dimension of the Hankel matrix tends to ∞ , and t tends to 0 + , such that s := 2n 2 t is finite, the double scaled Hankel determinant (effectively an operator determinant) has an integral representation in terms of a particular P III ′ . Expansions of the scaled Hankel determinant for small and large s are found. A further double scaling with α = −2n + λ, where n → ∞ and t, tends to 0 + , such that s := nt is finite. In this situation the scaled Hankel determinant has an integral representation in terms of a particular P V , and its small and large s asymptotic expansions are also found. The reproducing kernel in terms of monic polynomials orthogonal with respect to the Pollaczek-Jacobi type weight, under the origin (or hard edge) scaling may be expressed in terms of the solutions of a second order linear ordinary differential equation (ODE). * chenminfst@gmail.com † Corresponding author(Yang Chen), yayangchen@umac.mo and yangbrookchen@yahoo.co.uk ‡ faneg@fudan.edu.cn 1 With special choices of the parameters, the limiting (double scaled) kernel and the second order ODE degenerate to Bessel kernel and the Bessel differential equation, respectively. We also applied this method to polynomials orthogonal with respect to the perturbed Laguerre weight; w(x; t, α) := x α e −x e −t/x , 0 ≤ x < ∞, α > 0, t > 0. The scaled kernel at origin of this perturbed Laguerre ensemble has the same behavior with the above limiting kernel, although difference scaled schemes are adopted on these two kernels.2

show abstract

“…The weight w(x, α, t) = x α e −x−t/x was studied by Chen and Its [14] for finite n, Chen and Chen et al [8] for n → ∞.…”

Section: )mentioning

confidence: 99%

Orthogonal polynomials, asymptotics, and Heun equations

Chen

Filipuk

Zhan

2019

Journal of Mathematical Physics

Self Cite

View full text Add to dashboard Cite

The Painlevé equations arise from the study of Hankel determinants generated by moment matrices, whose weights are expressed as the product of "classical" weights multiplied by suitable "deformation factors", usually dependent on a "time variable" t. From ladder operators [12][13][14]30] one finds second order linear ordinary differential equations for associated orthogonal polynomials with coefficients being rational functions. The Painlevé and related functions appear as the residues of these rational functions.We will be interested in the situation when n, the order of the Hankel matrix and also the degree of the polynomials P n (x) orthogonal with respect to the deformed weights, gets large. We show that the second order linear differential equations satisfied by P n (x) are particular cases of Heun equations when n is large. In some sense, monic orthogonal polynomials generated by deformed weights mentioned below are solutions of a variety of Heun equations. Heun equations are of considerable importance in mathematical physics and in the special cases they degenerate to the hypergeometric and confluent hypergeometric equations (see, for instance, [1,23,36]).In this paper we look at three type of weights: the Jacobi type, the Laguerre type and the weights deformed by the indicator function of (a, b) χ (a,b) and the step function θ(x).In particular, we consider the following Jacobi type weights: 1.

show abstract

Singular linear statistics of the Laguerre unitary ensemble and Painlevé. III. Double scaling analysis

Cited by 26 publications

References 40 publications

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

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

Perturbed Hankel determinant, correlation functions and Painlevé equations

Orthogonal polynomials, asymptotics, and Heun equations

Contact Info

Product

Resources

About