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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 (a,a)(0<a<1) 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 Hn(a), Rn(a), and rn(a). 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 Hn(a) can be reduced to the Jimbo–Miwa–Okamoto σ form of the PainlevĂ© V equation.
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Document Type: Research Article

Publication date: February 1, 2018

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