Browsing by Author "Gharakhloo, Roozbeh"

Now showing 1 - 4 of 4
  • Thumbnail Image
    Item
    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.
  • Thumbnail Image
    Item
    Asymptotics of bordered Toeplitz determinants and next-to-diagonal Ising correlations
    (arXiv, 2021) Basor, Estelle; Ehrhardt, Torsten; Gharakhloo, Roozbeh; Its, Alexander; Li, Yuqi; Mathematical Sciences, School of Science
    We prove the analogue of the strong Szeg{\H o} limit theorem for a large class of bordered Toeplitz determinants. In particular, by applying our results to the formula of Au-Yang and Perk \cite{YP} for the next-to-diagonal correlations ⟨σ0,0σN−1,N⟩ in the anisotropic square lattice Ising model, we rigorously justify that the next-to-diagonal long-range order is the same as the diagonal and horizontal ones in the low temperature regime. The anisotropy-dependence of the subleading term in the asymptotics of the next-to-diagonal correlations is also established. We use Riemann-Hilbert and operator theory techniques, independently and in parallel, to prove these results.
  • Thumbnail Image
    Item
    A Riemann-Hilbert Approach to Asymptotic Analysis of Toeplitz+Hankel Determinants
    (SIGMA. Symmetry, Integrability and Geometry: Methods and Applications, 2020-10-06) Gharakhloo, Roozbeh; Its, Alexander; Mathematical Sciences, School of Science
    In this paper we will formulate 4×4 Riemann-Hilbert problems for Toeplitz+Hankel determinants and the associated system of orthogonal polynomials, when the Hankel symbol is supported on the unit circle and also when it is supported on an interval [a,b], 0
  • Thumbnail Image
    Item
    Riemann–Hilbert approach to a generalized sine kernel
    (Springer Link, 2020-02-01) Gharakhloo, Roozbeh; Its, Alexander R.; Kozlowski, Karol K.; Mathematical Sciences, School of Science
    We derive the large-distance asymptotics of the Fredholm determinant of the so-called generalized sine kernel at the critical point. This kernel corresponds to a generalization of the pure sine kernel arising in the theory of random matrices and has potential applications to the analysis of the large-distance asymptotic behaviour of the so-called emptiness formation probability for various quantum integrable models away from their free fermion point.