The smallest eigenvalue of large Hankel matrices generated by a singularly perturbed Laguerre weight

@article{Zhu2020TheSE,
  title={The smallest eigenvalue of large Hankel matrices generated by a singularly perturbed Laguerre weight},
  author={Mengkun Zhu and Yang Chen and Chuanzhong Li},
  journal={arXiv: Mathematical Physics},
  year={2020}
}
An asymptotic expression of the orthonormal polynomials $\mathcal{P}_{N}(z)$ as $N\rightarrow\infty$, associated with the singularly perturbed Laguerre weight $w_{\alpha}(x;t)=x^{\alpha}{\rm e}^{-x-\frac{t}{x}},~x\in[0,\infty),~\alpha>-1,~t\geq0$ is derived. Based on this, we establish the asymptotic behavior of the smallest eigenvalue, $\lambda_{N}$, of the Hankel matrix generated by the weight $w_{\alpha}(x;t)$. 

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SHOWING 1-10 OF 27 REFERENCES
The smallest eigenvalue of large Hankel matrices
The smallest eigenvalue of large Hankel matrices generated by a deformed Laguerre weight
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The Smallest Eigenvalue of Hankel Matrices
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Small eigenvalues of large Hankel matrices
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Singular linear statistics of the Laguerre unitary ensemble and Painlevé. III. Double scaling analysis
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Computing the smallest eigenvalue of large ill-conditioned Hankel matrices
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a(x) on the finite interval [a, b], rb cn= fXn da(x) a where w(x) = a'(x) satisfies rb log w(x) ?~~~~ dx > - oo. (x - a)112(b -X)12
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