Singular linear statistics of the Laguerre Unitary Ensemble and Painlevé III (PIII): Double scaling analysis

الأصل · EN

We continue with the study of the Hankel determinant, Dₙ(t,α):=(∫₀∞xʲ⁺ᵏw(x;t,α)dx)ⱼ,ₖ₌₀ⁿ⁻¹, generated by singularly perturbed Laguerre weight, w(x;t,α):=xα e⁻ˣ e⁻ᵗ/ˣ, 0≤ x<∞,α>0,t>0, obtained through a deformation of the Laguerre weight function, w(x;0,α):=xα e⁻ˣ, 0≤ x<∞, α>0, via the multiplicative factor e⁻ᵗ/ˣ. An earlier investigation was made on the finite n aspect of the problem, this has appeared in ci1. There, it was found that the logarithm of the Hankel determinant has an integral representation in terms of a particular PIII, 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 ∞, 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 PIII, albeit with fewer number of parameters. Expansions of the double scaled determinant for small and large parameter are obtained.