On the ratio probability of the smallest eigenvalues in the Laguerre unitary ensemble
(2018) Nonlinearity — Vol. 31, n° 4, p. 1155-1196 (2018)
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- Abstract
- 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 $x$.
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Atkin, M. R., Charlier, C., & Zohren, S. (2018). On the ratio probability of the smallest eigenvalues in the Laguerre unitary ensemble. Nonlinearity, 31(4), 1155-1196. https://doi.org/10.1088/1361-6544/aa9d57 (Original work published 2018)