Singular linear statistics of the Laguerre Unitary Ensemble and Painlev\'e III (${\mathrm{P}}_{\mathrm{I}\mathrm{I}\mathrm{I}}$): Double scaling analysis

We continue with the study of the Hankel determinant, $$D_n(t,\alpha ):=\det {({\int }_0^{\mathrm{\infty }}{x}^{j+k}w(x;t,\alpha )dx)}_{j,k=0}^{n-1},$$ generated by singularly perturbed Laguerre weight, $$w(x;t,\alpha ):=x^{\alpha }{\mathrm{e}}^{-x}{\mathrm{e}}^{-t\mathrm{/}x},0\le x<\mathrm{\infty },\alpha >0,t>0,$$ obtained through a deformation of the Laguerre weight function, $$w(x;0,\alpha ):=x^{\alpha }{\mathrm{e}}^{-x},0\le x<\mathrm{\infty },\alpha >0,$$ via the multiplicative factor ${\mathrm{e}}^{-t\mathrm{/}x}$. \ An earlier investigation was made on the finite $n$ aspect of the problem, this has appeared in \cite{ci1}. There, it was found that the logarithm of the Hankel determinant has an integral representation in terms of a particular ${\mathrm{P}}_{\mathrm{I}\mathrm{I}\mathrm{I}},$ and its derivative with $t\mathrm{.}$ In this paper we show that, under a double scaling, where $n$, the order of the Hankel matrix tends to $\mathrm{\infty },$ and $t$, tends to $0$, the scaled—and therefore, in some sense, infinite dimensional—Hankel determinant, has an integral representation in terms of the $C$ potential, and its derivatives. The second order non-linear differential equation which the $C$ potential satisfies, after a minor change of variables, is another ${\mathrm{P}}_{\mathrm{I}\mathrm{I}\mathrm{I}},$ albeit with fewer number of parameters. \ Expansions of the double scaled determinant for small and large parameter are obtained.