  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 $x$.
