Painlevé V and the Hankel determinant for a singularly perturbed Jacobi weight

Abstract

We study the Hankel determinant generated by a singularly perturbed Jacobi weightw(x,t):=(1−x2)αe−tx2,x∈[−1,1],α>0,t≥0. If t=0, it is reduced to the classical symmetric Jacobi weight. For t>0, the factor e−tx2 induces an infinitely strong zero at the origin. This Hankel determinant is related to the Wigner time-delay distribution in chaotic cavities.In the finite n dimensional case, we obtain two auxiliary quantities Rn(t) and rn(t) by using the ladder operator approach. We show that the Hankel determinant has an integral representation in terms of Rn(t), where Rn(t) is closely related to a particular Painlevé V transcendent. Furthermore, we derive a second-order nonlinear differential equation and also a second-order difference equation for the logarithmic derivative of the Hankel determinant. This quantity can be expressed in terms of the Jimbo-Miwa-Okamoto σ-function of a particular Painlevé V. Then we consider the asymptotics of the Hankel determinant under a suitable double scaling, i.e. n→∞ and t→0 such that s=2n2t is fixed. Based on previous results by using the Coulomb fluid method, we obtain the large s and small s asymptotic behaviors of the scaled Hankel determinant, including the constant term in the asymptotic expansion.