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 $n\to \mathrm{\infty }$ is found as an integral over $(0,\mathrm{\infty })$ 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 $(r-1)x\to \mathrm{\infty }$, as well large $n$ asymptotics for the associated Hankel determinants in several regimes of $r$ and $x$.