On the ratio probability of the smallest eigenvalues in the Laguerre Unitary Ensemble

We study the probability distribution of the ratio between the second smallest and smallest eigenvalue in the $n\times n$ Laguerre Unitary Ensemble. The probability that this ratio is greater than $r>1$ is expressed in terms of an $n \times n$ Hankel determinant with a perturbed Laguerre weight. The limiting probability distribution for the ratio as $n\to\infty$ is found as an integral over $(0,\infty)$ containing two functions $q_{1}(x)$ and $q_{2}(x)$. These functions satisfy a system of two coupled Painlev\'e V equations, which are derived from a Lax pair of a Riemann-Hilbert problem. We compute asymptotic behaviours of these functions as $rx \to 0_{+}$ and $(r-1)x \to \infty$, as well large $n$ asymptotics for the associated Hankel determinants in several regimes of $r$ and $x$.