Discrete, Continuous and Asymptotic for a Modified Singularly Gaussian Unitary Ensemble and the Smallest Eigenvalue of Its Large Hankel Matrices

This paper focuses on the characteristics of the Hankel determinant generated by a modified singularly Gaussian weight. The weight function is defined as:

$$\begin{aligned} w(z;t)=|z|^{\alpha }\textrm{e}^{-\frac{1}{z^2}-t\left( z^2-\frac{1}{z^2}\right) }, ~z\in {\mathbb {R}}, \end{aligned}$$

where \(\alpha >1\) and \(t\in (0,1)\) are parameters. Using ladder operator techniques, we derive a series of difference formulas for the auxiliary quantities and recurrence coefficients. We present the difference equations for the recurrence coefficients and the logarithmic derivative of the Hankel determinant. We then use the “t-dependence" to obtain the differential identities satisfied by the auxiliary quantities and the logarithmic derivative of the Hankel determinant. To obtain the large n asymptotic expressions of the recurrence coefficients, we use the Coulomb fluid method together with the related difference equations, which depend on n either being odd or even. We also obtain the reduction forms of the second-order differential equations satisfied by the orthogonal polynomials generated by this weight. Two special cases coincide with the bi-confluent Heun equation and the double confluent Heun equation, respectively. Finally, we calculate the asymptotic behavior of the smallest eigenvalue of large Hankel matrices generated by this weight. Our result not only covers the classical result of Szegö (Trans Am Math Soc 40:450–461, 1936) but also determines our next research direction.