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

Abstract

We continue with the study of the Hankel determinant, $$D_n(t,\alpha ):=\det {\left({\int }_0^{\infty }{x}^{j+k}w(x;t,\alpha )dx\right)}_{j,k=0}^{n-1},$$ generated by singularly perturbed Laguerre weight, $$w(x;t,\alpha ):=x^{\alpha }{\mathrm{e}}^{-x}{\mathrm{e}}^{-t/x},0\le x<\infty ,\alpha >0,t>0,$$ obtained through a deformation of the Laguerre weight function, $$w(x;0,\alpha ):=x^{\alpha }{\mathrm{e}}^{-x},0\le x<\infty ,\alpha >0,$$ via the multiplicative factor ${\mathrm{e}}^{-t/x}$. \\ 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 ${\mathrm{P}}_{\mathrm{III}},$ and its derivative with $t.$ In this paper we show that, under a double scaling, where $n$, the order of the Hankel matrix tends to $\infty ,$ 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 ${\mathrm{P}}_{\mathrm{III}},$ albeit with fewer number of parameters. \\ Expansions of the double scaled determinant for small and large parameter are obtained.