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Item Asymptotic Analysis of Structured Determinants via the Riemann-Hilbert Approach(2019-08) Gharakhloo, Roozbeh; Its, Alexander; Bleher, Pavel; Yattselev, Maxim; Eremenko, AlexandreIn 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.Item Lee–Yang zeros for the DHL and 2D rational dynamics, I. Foliation of the physical cylinder(Elsevier, 2017-05) Bleher, Pavel; Lyubich, Mikhail; Roeder, Roland; Department of Mathematical Sciences, School of ScienceIn a classical work of the 1950's, Lee and Yang proved that the zeros of the partition functions of a ferromagnetic Ising model always lie on the unit circle. Distribution of these zeros is physically important as it controls phase transitions in the model. We study this distribution for the Migdal–Kadanoff Diamond Hierarchical Lattice (DHL). In this case, it can be described in terms of the dynamics of an explicit rational function R in two variables (the renormalization transformation). We prove that R is partially hyperbolic on an invariant cylinder C. The Lee–Yang zeros are organized in a transverse measure for the central-stable foliation of R|C. Their distribution is absolutely continuous. Its density is C∞ (and non-vanishing) below the critical temperature. Above the critical temperature, it is C∞ on a open dense subset, but it vanishes on the complementary set of positive measure.