Browsing by Author "Eremenko, Alexandre"

Now showing 1 - 8 of 8
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    Asymptotic Analysis of Structured Determinants via the Riemann-Hilbert Approach
    (2019-08) Gharakhloo, Roozbeh; Its, Alexander; Bleher, Pavel; Yattselev, Maxim; Eremenko, Alexandre
    In this work we use and develop Riemann-Hilbert techniques to study the asymptotic behavior of structured determinants. In chapter one we will review the main underlying definitions and ideas which will be extensively used throughout the thesis. Chapter two is devoted to the asymptotic analysis of Hankel determinants with Laguerre-type and Jacobi-type potentials with Fisher-Hartwig singularities. In chapter three we will propose a Riemann-Hilbert problem for Toeplitz+Hankel determinants. We will then analyze this Riemann-Hilbert problem for a certain family of Toeplitz and Hankel symbols. In Chapter four we will study the asymptotics of a certain bordered-Toeplitz determinant which is related to the next-to-diagonal correlations of the anisotropic Ising model. The analysis is based upon relating the bordered-Toeplitz determinant to the solution of the Riemann-Hilbert problem associated to pure Toeplitz determinants. Finally in chapter ve we will study the emptiness formation probability in the XXZ-spin 1/2 Heisenberg chain, or equivalently, the asymptotic analysis of the associated Fredholm determinant.
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    Asymptotics of the Fredholm determinant corresponding to the first bulk critical universality class in random matrix models
    (2013-11-06) Bothner, Thomas Joachim; Its, Alexander R.; Bleher, Pavel, 1947-; Tarasov, Vitaly; Eremenko, Alexandre; Mukhin, Evgeny
    We study the one-parameter family of determinants $det(I-\gamma K_{PII}),\gamma\in\mathbb{R}$ of an integrable Fredholm operator $K_{PII}$ acting on the interval $(-s,s)$ whose kernel is constructed out of the $\Psi$-function associated with the Hastings-McLeod solution of the second Painlev\'e equation. In case $\gamma=1$, this Fredholm determinant describes the critical behavior of the eigenvalue gap probabilities of a random Hermitian matrix chosen from the Unitary Ensemble in the bulk double scaling limit near a quadratic zero of the limiting mean eigenvalue density. Using the Riemann-Hilbert method, we evaluate the large $s$-asymptotics of $\det(I-\gamma K_{PII})$ for all values of the real parameter $\gamma$.
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    Connection Problem for Painlevé Tau Functions
    (2019-08) Prokhorov, Andrei; Its, Alexander; Bleher, Pavel; Eremenko, Alexandre; Tarasov, Vitaly
    We derive the differential identities for isomonodromic tau functions, describing their monodromy dependence. For Painlev´e equations we obtain them from the relation of tau function to classical action which is a consequence of quasihomogeneity of corresponding Hamiltonians. We use these identities to solve the connection problem for generic solution of Painlev´e-III(D8) equation, and homogeneous Painlev´e-II equation. We formulate conjectures on Hamiltonian and symplectic structure of general isomonodromic deformations we obtained during our studies and check them for Painlev´e equations.
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    The Dynamics of Twisted Tent Maps
    (2013-07-12) Chamblee, Stephen Joseph; Misiurewicz, Michał, 1948-; Roeder, Roland; Geller, William; Eremenko, Alexandre; Mukhin, Evgeny
    This paper is a study of the dynamics of a new family of maps from the complex plane to itself, which we call twisted tent maps. A twisted tent map is a complex generalization of a real tent map. The action of this map can be visualized as the complex scaling of the plane followed by folding the plane once. Most of the time, scaling by a complex number will \twist" the plane, hence the name. The "folding" both breaks analyticity (and even smoothness) and leads to interesting dynamics ranging from easily understood and highly geometric behavior to chaotic behavior and fractals.
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    Fuchsian Equations with Three Non-Apparent Singularities
    (National Academy of Science of Ukraine, 2018) Eremenko, Alexandre; Tarasov, Vitaly; Mathematical Sciences, School of Science
    We show that for every second order Fuchsian linear differential equation E with n singularities of which n−3 are apparent there exists a hypergeometric equation H and a linear differential operator with polynomial coefficients which maps the space of solutions of H into the space of solutions of E. This map is surjective for generic parameters. This justifies one statement of Klein (1905). We also count the number of such equations E with prescribed singularities and exponents. We apply these results to the description of conformal metrics of curvature 1 on the punctured sphere with conic singularities, all but three of them having integer angles.
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    Metrics with conic singularities and spherical polygons
    (2014) Eremenko, Alexandre; Gabrielov, Andrei; Tarasov, Vitaly; Department of Mathematics, School of Science
    A spherical nn-gon is a bordered surface homeomorphic to a closed disk, with nn distinguished boundary points called corners, equipped with a Riemannian metric of constant curvature 11, except at the corners, and such that the boundary arcs between the corners are geodesic. We discuss the problem of classification of these polygons and enumerate them in the case that two angles at the corners are not multiples of ππ. The problem is equivalent to classification of some second order linear differential equations with regular singularities, with real parameters and unitary monodromy.
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    Metrics with four conic singularities and spherical quadrilaterals
    (AMS, 2016) Eremenko, Alexandre; Gabrielov, Andrei; Tarasov, Vitaly; Department of Mathematical Sciences, School of Science
    A spherical quadrilateral is a bordered surface homeomorphic to a closed disk, with four distinguished boundary points called corners, equipped with a Riemannian metric of constant curvature 1, except at the corners, and such that the boundary arcs between the corners are geodesic. We discuss the problem of classification of these quadrilaterals and perform the classification up to isometry in the case that two angles at the corners are multiples of π. The problem is equivalent to classification of Heun's equations with real parameters and unitary monodromy.
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    Spherical quadrilaterals with three non-integer angles
    (2016) Eremenko, Alexandre; Gabrielov, Andrei; Tarasov, Vitaly; Department of Mathematical Sciences, School of Science
    A spherical quadrilateral is a bordered surface homeomorphic to a closed disk, with four distinguished boundary points called corners, equipped with a Riemannian metric of constant curvature 11, except at the corners, and such that the boundary arcs between the corners are geodesic. We discuss the problem of classification of these quadrilaterals and perform the classification up to isometry in the case that one corner of a quadrilateral is integer (i.e., its angle is a multiple of ππ) while the angles at its other three corners are not multiples of ππ. The problem is equivalent to classification of Heun's equations with real parameters and unitary monodromy, with the trivial monodromy at one of its four singular point.