Painlevé III asymptotics of Hankel determinants for a singularly perturbed Laguerre weight

Abstract

In this paper, we consider the Hankel determinants associated with the singularly perturbed Laguerre weight $w(x)=x^{\alpha }{e}^{-x-t/x}$, $x\in (0,\infty )$, $t>0$ and $\alpha >0$. When the matrix size $n\to \infty$, we obtain an asymptotic formula for the Hankel determinants, valid uniformly for $t\in (0,d]$, $d>0$ fixed. A particular Painlevé III transcendent is involved in the approximation, as well as in the large-$n$ asymptotics of the leading coefficients and recurrence coefficients for the corresponding perturbed Laguerre polynomials. The derivation is based on the asymptotic results in an earlier paper of the authors, obtained by using the Deift-Zhou nonlinear steepest descent method.

arXiv comment21 pages, 3 figures, minor revision, references updated. arXiv admin note: text overlap with arXiv:1309.4354

References in corpus (8)

• Quantum mechanical time-delay matrix in chaotic scattering
• Wigner time-delay distribution in chaotic cavities and freezing transition
• Toeplitz determinants with merging singularities
• Painlevé V and a Pollaczek-Jacobi type orthogonal polynomials
• Critical edge behavior and the Bessel to Airy transition in the singularly perturbed Laguerre unitary ensemble
• A matrix model with a singular weight and Painleve' III
• Are Bosonic Replicas Faulty?
• On an average over the Gaussian Unitary Ensemble

Cited by in corpus (3)

• On the Rate of Convergence in the Central Limit Theorem for Linear Statistics of Gaussian, Laguerre, and Jacobi Ensembles
• A Riemann-Hilbert problem for equations of Painlevé type in the one matrix model with semi-classical potential
• Random matrix ensembles with singularities and a hierarchy of Painlevé III equations