UM ETheses Collection (澳門大學電子學位論文庫)
 Title

Double scaling in some random matrix ensembles
 English Abstract

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Perturbed Laguerre and Jacobi unitary ensembles generated by multiplying a factor e −t/x with the Laguerre weight x α e −x , x ∈ R +, α > 0, and the deformed Jacobi weight, x α (1−x) β , x ∈ [0, 1], α > 0, β > 0, separately. Mathematically, it introduces an infinitely strong zero on both weights at origin at x = 0. We are interested in the double scaling limits of Hankel determinants and correlation functions via both perturbed weights. In the first Chapter, we introduce some concepts and the structure of this thesis. In the second Chapter, we study the Hankel determinant, generated by a perturbed Laguerre weight, w(x;t, α) = x α e −x e −t/x, x ∈ R +, α > 0, t > 0. For finite n, the Hankel determinant, is expressed in terms of finite n Painleve III ( ´ PIII), which appeared in Chen and Its (2010). We show that, under a double scaling, where n, the size of the Hankel matrix tends to ∞, and t tends to 0 +, the scaled—and therefore, in some sense, infinite dimensional Hankel determinant, has an integral representation in terms of a lesser PIII. Expansions of the double scaled determinant are obtained. In the third Chapter, we study the Hankel determinant, via a PollaczekJacobi type weight (perturbed Jacobi weight), w(x;t, α, β) := x α (1 − x) β e −t/x, x ∈ [0, 1], α > 0, β > 0, t ≥ 0. It was shown in Chen and Dai (2010), for finite n, that the logarithmic derivative of this Hankel determinant satisfies a JimboMiwaOkamoto σform of PV. We show that, under a double scaling, where n the dimension of the Hankel matrix tends to ∞, and t tends to 0, such that s := 2n 2 t is finite, the double scaled Hankel determinant has an expression in terms of a particular PIII0. Expansions of the scaled Hankel determinant are found. Then in the same spirit, we also consider another double scaling with α = −2n + λ, where n → ∞, and t tends to 0, such that s := nt is finite. The scaled Hankel determinant has an integral representation in terms of a simper PV, and its asymptotic expansions are found. Two reproducing kernels in terms of monic polynomials orthogonal with respect to the PollaczekJacobi type weight, and the perturbed Laguerre weight, under the origin (or hard edge) scaling, have the same limiting behaviors, albeit with their scaling schemes are quite different. In the fourth Chapter, we make a conclusion of this thesis, and propose some open problems.
 Issue Date
 2016
 Author
 Chen, Min
 Faculty
 Faculty of Science and Technology
 Department:
 Department of Mathematics
 Degree
 Ph.D.
 Subject

Random matrices
Mathematics  Department of Mathematics
 Supervisor
 Chen Yang
 Files In This Item:

Fulltext (Intranet only)
 Location
 1/F Zone C
 Supervisor
 