Singular linear statistics of the Laguerre Unitary Ensemble and Painlevé III (${\rm P_{III}}$): 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, $$ w(x;t,\alpha):=x^{\alpha}{\rm e}^{-x}:{\rm e}^{-t/x}, \quad 0\leq x<\infty,\;\;\;\alpha>0,\;\;\;\;t>0, $$ obtained through a deformation of the Laguerre weight function, $$ w(x;0,\alpha):=x^{\alpha}{\rm e}^{-x},\quad 0\leq x<\infty,\;\; \alpha>0, $$ via the multiplicative factor ${\rm e}^{-t/x}$. \ 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 ${\rm P_{III}},$ 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 ${\rm P_{III}},$ albeit with fewer number of parameters. \ Expansions of the double scaled determinant for small and large parameter are obtained.