11,991 research outputs found
Asymptotic Gap Probability Distributions of the Gaussian Unitary Ensembles and Jacobi Unitary Ensembles
In this paper, we address a class of problems in unitary ensembles.
Specifically, we study the probability that a gap symmetric about 0, i.e.
is found in the Gaussian unitary ensembles (GUE) and the Jacobi
unitary ensembles (JUE) (where in the JUE, we take the parameters
). By exploiting the even parity of the weight, a doubling of the
interval to for the GUE, and , for the (symmetric) JUE,
shows that the gap probabilities maybe determined as the product of the
smallest eigenvalue distributions of the LUE with parameter and
and the (shifted) JUE with weights and
The function, namely, the derivative of the
log of the smallest eigenvalue distributions of the finite- LUE or the JUE,
satisfies the Jimbo-Miwa-Okamoto form of and ,
although in the shift Jacobi case, with the weight
the parameter does not show up in the equation. We also obtain the
asymptotic expansions for the smallest eigenvalue distributions of the Laguerre
unitary and Jacobi unitary ensembles after appropriate double scalings, and
obtained the constants in the asymptotic expansion of the gap probablities,
expressed in term of the Barnes function valuated at special point.Comment: 38 page
Painlev\'e III and the Hankel Determinant Generated by a Singularly Perturbed Gaussian Weight
In this paper, we study the Hankel determinant generated by a singularly
perturbed Gaussian weight By using the ladder operator approach associated with the orthogonal
polynomials, we show that the logarithmic derivative of the Hankel determinant
satisfies both a non-linear second order difference equation and a non-linear
second order differential equation. The Hankel determinant also admits an
integral representation involving a Painlev\'e III. Furthermore, we consider
the asymptotics of the Hankel determinant under a double scaling, i.e.
and such that is fixed. The
asymptotic expansions of the scaled Hankel determinant for large and small
are established, from which Dyson's constant appears.Comment: 22 page
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