A matrix model with a singular weight and Painleve' III

TL;DR

This paper analyzes a specific matrix model with a singular weight, revealing a novel phase transition described by Painleve' III in the double scaling limit, and provides explicit asymptotic formulas for the partition function.

Contribution

It is the first study to connect Painleve' III with double scaling limits in Random Matrix Theory, especially involving a singular weight and phase transition analysis.

Findings

Discovered a phase transition characterized by Painleve' III.

Computed asymptotics of the partition function in the double scaling limit.

Derived explicit initial conditions for the Painleve' III solution.

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 $(\sqrt{N} t, Nz^2 ) \to (u_1,u_2)$, 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 $O\bigl(N^{-1/2}\bigr)$. 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 $Nz^2\rightarrow u_2$ of this solution.