The Hankel Determinants from a Singularly Perturbed Jacobi Weight

Mathematics

Pengju Han¹  Yang Chen² 

[1] College of Science, Huazhong Agricultural University, Wuhan 430070, China;Department of Mathematics, Faculty of Science and Technology, University of Macau, Macau 999078, China;

关键词: random matrix theory;    Hankel determinant;    singularly perturbed Jacobi weight;    ladder operators;    Painlevé V;   

DOI  :  10.3390/math9222978

来源: DOAJ

【 摘 要 】

We study the Hankel determinant generated by a singularly perturbed Jacobi weight $w(x,s):={(1-x)}^{\alpha }{(1+x)}^{\beta }{\mathrm{e}}^{-\frac{s}{1-x}},x\in [-1,1],\alpha >0,\beta >0s\ge 0.$ If $s=0$, it is reduced to the classical Jacobi weight. For $s>0$, the factor ${\mathrm{e}}^{-\frac{s}{1-x}}$ induces an infinitely strong zero at $x=1$. For the finite n case, we obtain four auxiliary quantities $R_n\left(s\right)$, $r_n\left(s\right)$, ${\tilde{R}}_n\left(s\right)$, and ${\tilde{r}}_n\left(s\right)$ by using the ladder operator approach. We show that the recurrence coefficients are expressed in terms of the four auxiliary quantities with the aid of the compatibility conditions. Furthermore, we derive a shifted Jimbo–Miwa–Okamoto $\sigma$-function of a particular Painlevé V for the logarithmic derivative of the Hankel determinant $D_n\left(s\right)$. By variable substitution and some complicated calculations, we show that the quantity $R_n\left(s\right)$ satisfies the four Painlevé equations. For the large n case, we show that, under a double scaling, where n tends to ∞ and s tends to $0^{+}$, such that $\tau :=n^{2}s$ is finite, the scaled Hankel determinant can be expressed by a particular $P_{II{I}^{\prime }}$.

【 授权许可】

Unknown