Toeplitz Determinants Research Articles

Differential Subordination and Coefficient Functionals of Univalent Functions Related to $\cos z$

Differential subordination in the complex plane is the generalization of a differential inequality on the real line. In this paper, we consider two subclasses of univalent functions associated with the trigonometric function $\cos z$. Using some properties of the hypergeometric functions, we determine the sharp estimate on the parameter $\beta$ such that the analytic function $p(z)$ satisfying $p(0)=1$, is subordinate to $\cos z$ when the differential expression $p(z)+\beta z (dp(z)/dz)$ is subordinate to the Janowski function. We compute sharp bounds on coefficient functional Hermitian--Toeplitz determinants of the third and the fourth order with an invariance property for such functions. In addition, we estimate bound on Hankel determinants of the second and the third order.

Hankel and symmetric Toeplitz determinants for Sakaguchi starlike functions

Abstract. In this paper, we consider the class of starlike functions with respect to symmetric points which are also known as Sakaguchi starlike functions. We de- termine best possible bounds on Zalcman conjecture |a_n^2 – a_(2n-1) | and generalized Zalcman conjecture |aman − am+n−1| for n = 2 and n = 4, m = 2, respectively for such functions. Further, we compute estimate on third order and fourth order Hankel determinants. As well, we also obtain estimates on third and fourth symmetric Toeplitz determinants. Mathematics Subject Classification (2010): 30C45, 30C80. Keywords: Starlike function, Sakaguchi starlike functions, Zalcman conjecture, third and forth order Hankel determinants, second, third and fourth order symmetric Toeplitz determinants.

Asymptotics of the partition function of the perturbed Gross–Witten–Wadia unitary matrix model

AbstractWe consider the asymptotics of the partition function of the extended Gross–Witten–Wadia unitary matrix model by introducing an extra logarithmic term in the potential. The partition function can be written as a Toeplitz determinant with entries expressed in terms of the modified Bessel functions of the first kind and furnishes a ‐function sequence of the Painlevé equation. We derive the asymptotic expansions of the Toeplitz determinant up to and including the constant terms as the size of the determinant tends to infinity. The constant terms therein are expressed in terms of the Riemann zeta‐function and the Barnes ‐function. A third‐order phase transition in the leading terms of the asymptotic expansions is also observed.

Asymptotics of block Toeplitz determinants with piecewise continuous symbols

AbstractWe determine the asymptotics of the block Toeplitz determinants as for matrix‐valued piecewise continuous functions with a finitely many jumps under mild additional conditions. In particular, we prove that where , , and are constants that depend on the matrix symbol and are described in our main results. Our approach is based on a new localization theorem for Toeplitz determinants, a new method of computing the Fredholm index of Toeplitz operators with piecewise continuous matrix‐valued symbols, and other operator theoretic methods. As an application of our results, we consider piecewise continuous symbols that arise in the study of entanglement entropy in quantum spin chain models.

Toeplitz determinants in one and higher dimensions

Toeplitz determinants in one and higher dimensions

The Generalized Toeplitz Determinants for a Class of Holomorphic Mappings in Several Complex Variables

The Generalized Toeplitz Determinants for a Class of Holomorphic Mappings in Several Complex Variables

Hankel and Toeplitz Determinants of Logarithmic Coefficients of Inverse Functions for the Subclass of Starlike Functions with Respect to Symmetric Conjugate Points

This paper focuses on finding the upper bounds of the second Hankel and Toeplitz determinants, whose entries are logarithmic coefficients of inverse functions for a new subclass of starlike functions with respect to symmetric conjugate points associated with the exponential function defined by subordination. Results on initial Taylor coefficients and logarithmic coefficients of inverse functions for a new subclass are also presented. This study may inspire others to focus further to the coefficient functional problems associated with the inverse functions of various classes of univalent functions.

Boundary values of Hankel and Toeplitz determinants for [formula omitted]-convex functions

The study of holomorphic functions has been recently extended through the application of diverse techniques, among which quantum calculus stands out due to its wide-ranging applications across various scientific disciplines. In this context, we introduce a novel q-differential operator defined via the generalized binomial series, which leads to the derivation of new classes of quantum-convex (q-convex) functions. Several specific instances within these classes were explored in detail. Consequently, the boundary values of the Hankel determinants associated with these functions were analyzed. All graphical representations and computational analyses were performed using Mathematica 12.0.•These classes are defined by utilizing a new q-differential operator.•The coefficient values |ai|(i=2,3,4) are investigated.•Toeplitz determinants, such as the second T2(2) and the third T3(1) order inequalities, are calculated.

Coefficients Estimates for a Subclass of Starlike Functions

The paper mainly investigates the initial coefficients for the subclasses of starlike functions defined by using the Cosine function involving $\alpha$ ($0\leq\alpha<1$), we obtain upper bounds for initial order of Hankel determinants and symmetric Toeplitz determinants whose elements are the initial coefficients. Also, we obtain initial coefficient estimation of logarithmic coefficients for the subclass.

Symmetric Toeplitz Determinants for Classes Defined by Post Quantum Operators Subordinated to the Limaçon Function

The present extensive study is focused to find estimates for the upper bounds of the Toeplitz determinants. The logarithmic coefficients of univalent functions play an important role in different estimates in the theory of univalent functions, and in the this paper we derive the estimates of Toeplitz determinants and Toeplitz determinants of the logarithmic coefficients for the subclasses Ls𝑆𝑝𝑞 L𝑠C𝑝𝑞 and LsS𝑝𝑞 ∩ S, L𝑠𝐶𝑝𝑞 ∩ S, 0 q≤p≤10 𝑞≤𝑝≤1, defined by post quantum operators, which map the open unit disc 𝐷 onto the domain bounded by the limaçon curve defined by ∂Ds:={u+iv∈C:(u−1)2+v2−s4.2=4s2(u−1+s2)2+v2.}, where s∈−1,1.∖{0}. Keywords: Limaçon domain, subordination, (p, q)–derivative, Toeplitz and Hankel determinants, symmetric Toeplitz determinant, logarithmic coefficients, starlike functions with respect to symmetric points.

Certain sharp estimates of Ozaki close-to-convex functions

The goal of this paper is to establish the sharp estimates on coefficient functionals like Hermitian–Toeplitz determinant of second-order involving logarithmic coefficients, initial logarithmic inverse coefficients as well as sharp bounds on initial order Schwarzian derivatives of the Ozaki close-to-convex functions.

Multiplicative chaos measures from thick points of log‐correlated fields

AbstractWe prove that multiplicative chaos measures can be constructed from extreme level sets or thick points of the underlying logarithmically correlated field. We develop a method which covers the whole subcritical phase and only requires asymptotics of suitable exponential moments for the field. As an application, we establish that these estimates hold for the logarithm of the absolute value of the characteristic polynomial of a Haar distributed random unitary matrix (CUE), using known asymptotics for Toeplitz determinant with (merging) Fisher–Hartwig singularities. Hence, this proves a conjecture of Fyodorov and Keating concerning the fluctuations of the volume of thick points of the CUE characteristic polynomial.

Sharp Bounds on Toeplitz Determinants for Starlike and Convex Functions Associated with Bilinear Transformations

Sharp Bounds on Toeplitz Determinants for Starlike and Convex Functions Associated with Bilinear Transformations

Certain inequalities related with Hankel and Toeplitz determinant for q-starlike functions

In the present article, we prove the sharp upper bounds for the second order Hankel determinants |H2(1)|,|H2(2)| and related functionals |a2a3−a4|, |a2a5−a3a4| for q-starlike functions. An upper bound for the third order Hankel determinant |H3(1)| along with the sharp upper bounds for Toeplitz determinant |Tm(n)|, where (m,n)∈{(2,2),(2,3),(3,1),(3,2)} are attained. Many known results are also obtained as corollaries of our main results.

Exact asymptotics of long-range quantum correlations in a non-equilibrium steady state

Out-of-equilibrium states of many-body systems tend to evade a description by standard statistical mechanics, and their uniqueness is epitomized by the possibility of certain long-range correlations that cannot occur in equilibrium. In quantum many-body systems, coherent correlations of this sort may lead to the emergence of remarkable entanglement structures. In this work, we analytically study the asymptotic scaling of quantum correlation measures—the mutual information (MI) and the fermionic negativity—within the zero-temperature steady state of voltage-biased free fermions on a one-dimensional lattice containing a non-interacting impurity. Previously, we have shown that two subsystems on opposite sides of the impurity exhibit volume-law entanglement, which is independent of the absolute distances of the subsystems from the impurity. Here, we go beyond that result and derive the exact form of the subleading logarithmic corrections to the extensive terms of correlation measures, in excellent agreement with numerical calculations. In particular, the logarithmic term of the MI asymptotics can be encapsulated in a concise formula, depending only on simple four-point ratios of subsystem length scales and on the impurity scattering probabilities at the Fermi energies. This echoes the case of equilibrium states, where such logarithmic terms may convey universal information about the physical system. To compute these exact results, we devise a hybrid method that relies on Toeplitz determinant asymptotics for correlation matrices in both real space and momentum space, successfully circumventing the inhomogeneity of the system. This method could potentially find wider use for analytical calculations of entanglement measures in similar scenarios.

Generalised unitary group integrals of Ingham-Siegel and Fisher-Hartwig type

We generalise well-known integrals of Ingham-Siegel and Fisher-Hartwig type over the unitary group U(N) with respect to Haar measure, for finite N and including fixed external matrices. When depending only on the eigenvalues of the unitary matrix, such integrals can be related to a Toeplitz determinant with jump singularities. After introducing fixed deterministic matrices as external sources, the integrals can no longer be solved using Andréiéf’s integration formula. Resorting to the character expansion as put forward by Balantekin, we derive explicit determinantal formulae containing Kummer’s confluent and Gauß’ hypergeometric function. They depend only on the eigenvalues of the deterministic matrices and are analytic in the parameters of the jump singularities. Furthermore, unitary two-matrix integrals of the same type are proposed and solved in the same manner. When making part of the deterministic matrices random and integrating over them, we obtain similar formulae in terms of Pfaffian determinants. This is reminiscent to a unitary group integral found recently by Kanazawa and Kieburg [J. Phys. A: Math. Theor. 51(34), 345202 (2018)].

Toeplitz Determinant for the Inverse of a Function Whose Derivative Has a Positive Real Part

Abstract Let R denote the family of analytic and univalent functions f(z) in the unit disk Ω = {z ∈ C: |z | < 1} such that Re f′(z) > 0. We investigated the Toeplitz determinant for the inverse of the function to f ∈ R and gave the upper bound estimates of the determinants T 2(2), T 2(3), T 3(1) and T 3(2) .

Successive coefficients and Toeplitz determinant for concave univalent functions

Successive coefficients and Toeplitz determinant for concave univalent functions

Hankel and Toeplitz determinants for a subclass of analytic functions

Let the function $f\left( z \right) =z+\sum_{k=2}^{\infty}a{_{k}}z {^{k}}\in A$ be locally univalent for $z \in \mathbb{D}%:=\{z \in \mathbb{C}:{|}z {|}<1\}$ and $0\leq\alpha<1$.Then, $f$\textit{\ }$\in $ $M(\alpha )$ if and only if \begin{equation*}\Re\Big( \left( 1-z ^{2}\right) \frac{f(z )}{z }\Big) >\alpha,\quad z \in \mathbb{D}.\end{equation*}%Due to their geometrical characteristics, this class has a significantimpact on the theory of geometric functions. In the article we obtain sharp bounds for the second Hankel determinant \begin{equation*}\left\vert H_{2}\left( 2\right) \left( f\right) \right\vert =\left\verta_{2}a_{4}-{a_{3}^{2}}\right\vert \end{equation*}and some Toeplitz determinants \begin{equation*}\left\vert {T}_{3}\left( 1\right) \left( f\right) \right\vert =\left\vert 1-2%{a_{2}^{2}}+2{a_{2}^{2}}a_{3}-{a_{3}^{2}}\right\vert,\ \\left\vert {T}_{3}\left( 2\right) \left( f\right) \right\vert =\left\vert {%a_{2}^{3}}-2a_{2}{a_{3}^{2}}+2{a_{3}^{2}}a_{4}-a_{2}{a_{4}^{2}}\right\vert \end{equation*}of a subclass of analytic functions $M(\alpha )$ in the open unit disk $%\mathbb{D}$.

Exact Correlations in Topological Quantum Chains

Although free-fermion systems are considered exactly solvable, they generically do not admit closed expressions for nonlocal quantities such as topological string correlations or entanglement measures. We derive closed expressions for such quantities for a dense subclass of certain classes of topological fermionic wires (classes BDI and AIII). Our results also apply to spin chains called generalised cluster models. While there is a bijection between general models in these classes and Laurent polynomials, restricting to polynomials with degenerate zeros leads to a plethora of exact results: (1) we derive closed expressions for the string correlation functions - the order parameters for the topological phases in these classes; (2) we obtain an exact formula for the characteristic polynomial of the correlation matrix, giving insight into ground state entanglement; (3) the latter implies that the ground state can be described by a matrix product state (MPS) with a finite bond dimension in the thermodynamic limit - an independent and explicit construction for the BDI class is given in a concurrent work [Phys. Rev. Res. 3 (2021), 033265, 26 pages, arXiv:2105.12143]; (4) for BDI models with even integer topological invariant, all non-zero eigenvalues of the transfer matrix are identified as products of zeros and inverse zeros of the aforementioned polynomial. General models in these classes can be obtained by taking limits of the models we analyse, giving a further application of our results. To the best of our knowledge, these results constitute the first application of Day's formula and Gorodetsky's formula for Toeplitz determinants to many-body quantum physics.