A matrix model with a singular weight and Painleve' III

Abstract

We investigate the matrix model with weight $w(x):=\exp (-z^2/2x^2+t/x-x^2/2)$ and unitary symmetry. and unitary symmetry. In particular we study the double scaling limit as $N\to \infty$ and $(\sqrt{N}t,N{z}^2)\to (u_1,u_2)$, where $N$ is the matrix dimension and the parameters $(u_1,u_2)$ remain finite. Using the Deift-Zhou steepest descent method we compute the asymptotics of the partition function when $z$ and $t$ are of order $O\left(N^{-1/2}\right)$. In this regime we discover a phase transition in the $(z,N)$-plane characterised by the Painleve' III equation. This is the first time that Painleve' III appears in studies of double scaling limits in Random Matrix Theory and is associated to the emergence of an essential singularity in the weighting function. The asymptotics of the partition function is expressed in terms of a particular solution of the Painleve' III equation. We derive explicitly the initial conditions in the limit $N{z}^2\to u_2$ of this solution.

50 pages and 6 figures. Minor corrections

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Cited by in corpus (19)

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• Differential, Difference and Asymptotic Relations for Pollaczek-Jacobi Type Orthogonal Polynomials and Their Hankel Determinants
• Painlevé V and the Hankel Determinant for a Singularly Perturbed Jacobi Weight
• Gaussian unitary ensembles with pole singularities near the soft edge and a system of coupled Painlevé XXXIV equations
• A singular-potential random matrix model arising in mean-field glassy systems
• Hankel Determinant and Orthogonal Polynomials for a Perturbed Gaussian Weight: from Finite $n$ to Large $n$ Asymptotics
• Painlevé III asymptotics of Hankel determinants for a singularly perturbed Laguerre weight
• Random matrix ensembles with singularities and a hierarchy of Painlevé III equations
• Critical edge behavior in the singularly perturbed Pollaczek-Jacobi type unitary ensemble
• A Riemann-Hilbert problem for equations of Painlevé type in the one matrix model with semi-classical potential
• Asymptotics of the Largest Eigenvalue Distribution of the Laguerre Unitary Ensemble
• Gap probability at the hard edge for random matrix ensembles with pole singularities in the potential
• Gaussian unitary ensemble with two jump discontinuities, PDEs and the coupled Painlevé II and IV systems
• Painlevé III asymptotics of Hankel determinants for a perturbed Jacobi weight
• Riemann-Hilbert Theory without local Parametrix Problems: Applications to Orthogonal Polynomials
• Hankel determinants for a singular complex weight and the first and third Painlevé transcendents
• Critical edge behavior in the perturbed Laguerre ensemble and the Painleve V transcendent
• Hankel determinants for a perturbed Laguerre weight and Painleve V equation