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Painléve III and a singular linear statistics in Hermitian random matrix ensembles I.
, 2008
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Painleve VI and Hankel Determinants for the Generalized Jacobi Weight
, 2009
"... We study the Hankel determinant of the generalized Jacobi weight (x − t) γ x α (1 − x) β for x ∈ [0,1] with α,β> 0, t < 0 and γ ∈ R. Based on the ladder operators for the corresponding monic orthogonal polynomials Pn(x), it is shown that the logarithmic derivative of Hankel determinant is char ..."
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We study the Hankel determinant of the generalized Jacobi weight (x − t) γ x α (1 − x) β for x ∈ [0,1] with α,β> 0, t < 0 and γ ∈ R. Based on the ladder operators for the corresponding monic orthogonal polynomials Pn(x), it is shown that the logarithmic derivative of Hankel determinant is characterized by a τfunction for the Painlevé VI system.
The Hilbert series of N = . . .
"... We present a novel approach for computing the Hilbert series of 4d N = 1 supersymmetric QCD with SO(Nc) and Sp(Nc) gauge groups. It is shown that such Hilbert series can be recast in terms of determinants of Hankel matrices. With the aid of results from random matrix theory, such Hankel determinan ..."
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Cited by 5 (1 self)
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We present a novel approach for computing the Hilbert series of 4d N = 1 supersymmetric QCD with SO(Nc) and Sp(Nc) gauge groups. It is shown that such Hilbert series can be recast in terms of determinants of Hankel matrices. With the aid of results from random matrix theory, such Hankel determinants can be evaluated both exactly and asymptotically. Several new results on Hilbert series for general numbers of colours and flavours are thus obtained in this paper. We show that the Hilbert series give rise to families of rational solutions, with palindromic numerators, to the Painleve ́ VI equations. Due to the presence of such Painleve ́ equations, there exist integrable Hamiltonian systems that describe the moduli spaces of SO(Nc) and Sp(Nc) SQCD. To each system, we explicitly state the corresponding Hamiltonian and family of elliptic curves. It turns out that such elliptic curves take the same form as the Seiberg–Witten curves for 4d N = 2 SU(2) gauge theory with 4 flavours.
Painlevé II in random matrix theory and related fields. arXiv:1210.3381
, 2012
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Vortices and Polynomials
, 2009
"... The relationship between point vortex dynamics and the properties of polynomials with roots at the vortex positions is discussed. Classical polynomials, such as the Hermite polynomials, have roots that describe the equilibria of identical vortices on the line. Stationary and uniformly translating vo ..."
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The relationship between point vortex dynamics and the properties of polynomials with roots at the vortex positions is discussed. Classical polynomials, such as the Hermite polynomials, have roots that describe the equilibria of identical vortices on the line. Stationary and uniformly translating vortex configurations with vortices of the same strength but positive or negative orientation are given by the zeros of the Adler–Moser polynomials, which arise in the description of rational solutions of the Kortewegde Vries equation. For quadupole background flow, vortex configurations are given by the zeros of polynomials expressed as wronskians of Hermite polynomials. Further new solutions are found in this case using the special polynomials arising the in the description of rational solutions of the fourth Painlevé equation.
Perturbed Hankel determinants: Applications to the information theory of MIMO wireless communications
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