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

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].

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
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