Browsing by Author "Yattselev, Maxim"
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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 Investigation of the two-cut phase region in the complex cubic ensemble of random matrices(AIP Publishing, 2022-06) Barhoumi, Ahmad; Bleher, Pavel; Deaño, Alfredo; Yattselev, Maxim; Mathematical Sciences, School of ScienceWe investigate the phase diagram of the complex cubic unitary ensemble of random matrices with the potential V(M) = −1/3(M^3) + tM, where t is a complex parameter. As proven in our previous paper [Bleher et al., J. Stat. Phys. 166, 784–827 (2017)], the whole phase space of the model, t ∈ C, is partitioned into two phase regions, Oone−cut and Otwo−cut, such that in Oone−cut the equilibrium measure is supported by one Jordan arc (cut) and in Otwo−cut by two cuts. The regions Oone−cut and Otwo−cut are separated by critical curves, which can be calculated in terms of critical trajectories of an auxiliary quadratic differential. In Bleher et al. [J. Stat. Phys. 166, 784–827 (2017)], the one-cut phase region was investigated in detail. In the present paper, we investigate the two-cut region. We prove that in the two-cut region, the endpoints of the cuts are analytic functions of the real and imaginary parts of the parameter t, but not of the parameter t itself (so that the Cauchy–Riemann equations are violated for the endpoints). We also obtain the semiclassical asymptotics of the orthogonal polynomials associated with the ensemble of random matrices and their recurrence coefficients. The proofs are based on the Riemann–Hilbert approach to semiclassical asymptotics of the orthogonal polynomials and the theory of S-curves and quadratic differentials.Item On Random Polynomials Spanned by OPUC(2020-12) Aljubran, Hanan; Yattselev, Maxim; Bleher, Pavel; Mukhin, Evgeny; Roeder, RolandWe consider the behavior of zeros of random polynomials of the from \begin{equation*} P_{n,m}(z) := \eta_0\varphi_m^{(m)}(z) + \eta_1 \varphi_{m+1}^{(m)}(z) + \cdots + \eta_n \varphi_{n+m}^{(m)}(z) \end{equation*} as \( n\to\infty \), where \( m \) is a non-negative integer (most of the work deal with the case \( m =0 \) ), \( \{\eta_n\}_{n=0}^\infty \) is a sequence of i.i.d. Gaussian random variables, and \( \{\varphi_n(z)\}_{n=0}^\infty \) is a sequence of orthonormal polynomials on the unit circle \( \mathbb T \) for some Borel measure \( \mu \) on \( \mathbb T \) with infinitely many points in its support. Most of the work is done by manipulating the density function for the expected number of zeros of a random polynomial, which we call the intensity function.Item Orthogonal Polynomials on S-Curves Associated with Genus One Surfaces(2020-08) Barhoumi, Ahmad; Yattselev, Maxim; Bleher, Pavel; Its, Alexander; Tarasov, VitalyWe consider orthogonal polynomials P_n satisfying orthogonality relations where the measure of orthogonality is, in general, a complex-valued Borel measure supported on subsets of the complex plane. In our consideration we will focus on measures of the form d\mu(z) = \rho(z) dz where the function \rho may depend on other auxiliary parameters. Much of the asymptotic analysis is done via the Riemann-Hilbert problem and the Deift-Zhou nonlinear steepest descent method, and relies heavily on notions from logarithmic potential theory.Item Topological Expansion in the Complex Cubic Log–Gas Model: One-Cut Case(2017-02) Bleher, Pavel; Deaño, Alfredo; Yattselev, Maxim; Department of Mathematical Sciences, School of ScienceWe prove the topological expansion for the cubic log–gas partition function ZN(t)=∫Γ⋯∫Γ ∏1≤j