# Gap Probability Distribution of the Jacobi Unitary Ensemble: An Elementary Treatment, from Finite n to Double Scaling

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In this paper, we study the gap probability problem of the (symmetric) Jacobi unitary ensemble of Hermitian random matrices, namely, the probability that the interval $\left(-a,a\right)\phantom{\rule{0.222222em}{0ex}}\left(0 is free of eigenvalues. Using the ladder operator technique for orthogonal polynomials and the associated supplementary conditions, we derive three quantities instrumental in the gap probability, denoted by ${H}_{n}\left(a\right)$, ${R}_{n}\left(a\right)$, and ${r}_{n}\left(a\right)$. We find that each one satisfies a second‐order differential equation. We show that after a double scaling, the large second‐order differential equation in the variable a with n as parameter satisfied by ${H}_{n}\left(a\right)$ can be reduced to the Jimbo–Miwa–Okamoto σ form of the Painlevé V equation.
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Document Type: Research Article

Publication date: February 1, 2018

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