# Painlev\'{e} V, Painlev\'{e} XXXIV and the Degenerate Laguerre Unitary Ensemble

## TL;DR

This paper investigates the Hankel determinant linked to the degenerate Laguerre unitary ensemble, revealing connections to Painlevé V and XXXIV equations through ladder operators, Riccati equations, and asymptotic analysis.

## Contribution

It derives new ladder operators and compatibility conditions for the degenerate Laguerre ensemble, establishing links to Painlevé V and XXXIV equations and analyzing large n asymptotics.

## Key findings

- R_n(t) satisfies Painlevé V equation.
- σ_n(t) obeys Jimbo-Miwa-Okamoto σ-form of Painlevé V.
- Large n asymptotics lead to Painlevé XXXIV equation.

## Abstract

In this paper, we study the Hankel determinant associated with the degenerate Laguerre unitary ensemble. This problem originates from the largest or smallest eigenvalue distribution of the degenerate Laguerre unitary ensemble. We derive the ladder operators and its compatibility condition with respect to a general perturbed weight. By applying the ladder operators to our problem, we obtain two auxiliary quantities $R_n(t)$ and $r_n(t)$ and show that they satisfy the coupled Riccati equations, from which we find that $R_n(t)$ satisfies the Painlev\'{e} V equation. Furthermore, we prove that $\sigma_{n}(t)$, a quantity related to the logarithmic derivative of the Hankel determinant, satisfies both the continuous and discrete Jimbo-Miwa-Okamoto $\sigma$-form of the Painlev\'{e} V. In the end, by using Dyson's Coulomb fluid approach, we consider the large $n$ asymptotic behavior of our problem at the soft edge, which gives rise to the Painlev\'{e} XXXIV equation.