Hankel determinant and orthogonal polynomials for the Gaussian weight with a jump. (English) Zbl 1163.15027

Its, A.; Krasovsky, I.

Baik, Jinho (ed.) et al., Integrable systems and random matrices. In honor of Percy Deift. Conference on integrable systems, random matrices, and applications in honor of Percy Deift’s 60th birthday, New York, NY, USA, May 22–26, 2006. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4240-9/pbk). Contemporary Mathematics 458, 215-247 (2008).

The authors study asymptotics of Hankel determinants
\[ D_n(\beta)=\det\left(\int_{-\infty}^\infty x^{j+k}w(x)\,dx\right)_{j,k=0}^{n-1} \quad(n\to\infty) \]
with discontinuous symbol given by \(w(x)=e^{-x^2}e^{i\beta\pi}\) for \(x<\mu_0\) and \(w(x)=e^{-x^2}e^{-i\beta\pi}\) for \(x\geq\mu_0\), where \(\text{Re}\,\beta(-1/4,1/4)\). Fix \(\lambda_0\in(-1,1)\) and let \(\mu_0=\lambda_0\sqrt{2n}\).
The main result of the paper is the asymptotic formula
\[ \begin{split} {D_n(\beta)\over D_n(0)}=G(1+\beta)G(1-\beta)(1-\lambda_0^2)^{-3\beta^2/2}(8n)^{-\beta^2}\\ \times \exp\left\{2in\beta\left(\arcsin\lambda_0+\lambda_0\sqrt{1-\lambda_0^2}\right)\right\} \left[1+O\left( {\ln n\over n^{1-4| \text{ Re}\,\beta| }} \right)\right] \quad(n\to\infty), \end{split} \]
where \(G(z)\) is the Barnes \(G\)-function. The proof is based on the Riemann-Hilbert analysis of a related system of orthogonal polynomials.
For the entire collection see [Zbl 1139.37001].

Reviewer: Alexei Yu. Karlovich (Lisboa)

MSC:

15B52	Random matrices (algebraic aspects)
47B35	Toeplitz operators, Hankel operators, Wiener-Hopf operators
34E05	Asymptotic expansions of solutions to ordinary differential equations
34M50	Inverse problems (Riemann-Hilbert, inverse differential Galois, etc.) for ordinary differential equations in the complex domain
42C05	Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
15A15	Determinants, permanents, traces, other special matrix functions

Keywords:

Hankel determinant; asymptotic formula; Fisher-Hartwig singularity; Barnes \(G\)-function; Riemann-Hilbert problem; orthogonal polynomials; random matrices