Fisher-Hartwig Singularity Research Articles

On the Fredholm determinant of the confluent hypergeometric kernel with discontinuities

We consider the determinantal point process with the confluent hypergeometric kernel. This process is a universal point process in random matrix theory and describes the distribution of eigenvalues of large random Hermitian matrices near the Fisher-Hartwig singularity. Applying the Riemann-Hilbert method, we study the generating function of this process on any given number of intervals. It can be expressed as the Fredholm determinant of the confluent hypergeometric kernel with n discontinuities. In this paper, we derive an integral representation for the determinant by using the Hamiltonian of the coupled Painlevé V system. By evaluating the total integral of the Hamiltonian, we obtain the asymptotics of the determinant as the n discontinuities tend to infinity up to and including the constant term. Here the constant term is expressed in terms of the Barnes G-function.

Cyclic Pólya Ensembles on the Unitary Matrices and their Spectral Statistics

A framework to study the eigenvalue probability density function for products of unitary random matrices with an invariance property is developed. This involves isolating a class of invariant unitary matrices, to be referred to as cyclic Pólya ensembles, and examining their properties with respect to the spherical transform on $$\mathrm U(N)$$ . Included in the cyclic Pólya ensemble class are Haar invariant unitary matrices, the circular Jacobi ensemble, known in relation to the Fisher-Hartwig singularity in the theory of Toeplitz determinants, as well as the heat kernel for Brownian motion on the unitary group. We define cyclic Pólya frequency functions and show their relation to the cyclic Pólya ensembles, and give a uniqueness statement for the corresponding weights. The natural appearance of bilateral hypergeometric series is highlighted, with this special function playing the role of the Meijer G-function in the transform theory of unitary invariant product of positive definite matrices. We construct a family of functions forming bi-orthonormal pairs which underly the correlation kernel of the corresponding determinantal point processes, and furthermore obtain an integral formula for the correlation kernel involving just two of these functions.

Crossing bridges with strong Szeg\u0151 limit theorem

We develop a new technique for computing a class of four-point correlation functions of heavy half-BPS operators in planar mathcal{N} = 4 SYM theory which admit factorization into a product of two octagon form factors with an arbitrary bridge length. We show that the octagon can be expressed as the Fredholm determinant of the integrable Bessel operator and demonstrate that this representation is very efficient in finding the octagons both at weak and strong coupling. At weak coupling, in the limit when the four half-BPS operators become null separated in a sequential manner, the octagon obeys the Toda lattice equations and can be found in a closed form. At strong coupling, we exploit the strong Szegő limit theorem to derive the leading asymptotic behavior of the octagon and, then, apply the method of differential equations to determine the remaining subleading terms of the strong coupling expansion to any order in the inverse coupling. To achieve this goal, we generalize results available in the literature for the asymptotic behavior of the determinant of the Bessel operator. As a byproduct of our analysis, we formulate a Szegő-Akhiezer-Kac formula for the determinant of the Bessel operator with a Fisher-Hartwig singularity and develop a systematic approach to account for subleading power suppressed contributions.

Multiple phases in a generalized Gross-Witten-Wadia matrix model

We study a unitary matrix model of the Gross-Witten-Wadia type, extended with the addition of characteristic polynomial insertions. The model interpolates between solvable unitary matrix models and is the unitary counterpart of a deformed Cauchy ensemble. Exact formulas for the partition function and Wilson loops are given in terms of Toeplitz determinants and minors and large N results are obtained by using Szegö theorem with a Fisher-Hartwig singularity. In the large N (planar) limit with two scaled couplings, the theory exhibits a surprisingly intricate phase structure in the two-dimensional parameter space.

Gap Probability of the Circular Unitary Ensemble with a Fisher–Hartwig Singularity and the Coupled Painlevé V System

We consider the circular unitary ensemble with a Fisher-Hartwig singularity of both jump type and root type at $z=1$. A rescaling of the ensemble at the Fisher-Hartwig singularity leads to the confluent hypergeometric kernel. By studying the asymptotics of the Toeplitz determinants, we show that the probability of there being no eigenvalues in a symmetric arc about the singularity on the unit circle for a random matrix in the ensemble can be explicitly evaluated via an integral of the Hamiltonian of the coupled Painlev\'{e} V system in dimension four. This leads to a Painlev\'e-type representation of the confluent hypergeometric-kernel determinant. Moreover, the large gap asymptotics, including the constant terms, are derived by evaluating the total integral of the Hamiltonian. In particular, we reproduce the large gap asymptotics of the confluent hypergeometric-kernel determinant obtained by Deift, Krasovsky and Vasilevska, and the sine-kernel determinant as a special case, including the constant term conjectured earlier by Dyson.

Phase transition in complex-time Loschmidt echo of short and long range spin chain

We explain and exploit the random matrix formulation of the Loschmidt echo for the XX spin chain, valid for multiple domain wall initial states and also for an XX spin chain generalized with additional interactions to more neighbours. For models with interactions decaying as , with p integer or natural number and α ⩾ 0, we show that there are third order phase transitions in a scaling limit of the complex-time Loschmidt echo amplitudes. For the long-range version of the chain, we use an exact result for Toeplitz determinants with a pure Fisher–Hartwig singularity, to obtain exactly the Loschmidt echo for complex times and discuss the associated Stokes phenomena. We also study the case of a finite chain for one of the generalized XX models.

Transition asymptotics of Toeplitz determinants and emergence of Fisher–Hartwig representations

We compute the transition asymptotics (double-scaling limits) of Toeplitz determinants generated by symbols f t possessing Fisher–Hartwig singularities. The symbols f t that we consider depend on a parameter t such that f t has one Fisher–Hartwig singularity when t > 0 and two Fisher–Hartwig singularities when t = 0. Unlike in the other studies of the transition asymptotics of Toeplitz determinants, our setting involves the emergence of Fisher–Hartwig representations as . We use the Riemann–Hilbert problem for orthogonal polynomials and its connection to Painlevé transcendents to obtain the asymptotics. We apply our results to study a special correlator known as the emptiness formation probability (EFP) for the one-dimensional anisotropic XY spin-1/2 chain in a transverse magnetic field, and describe its transition between different regions in the phase diagram across critical lines.

On the moments of the characteristic polynomial of a Ginibre random matrix

In this article we study the large $N$ asymptotics of complex moments of the absolute value of the characteristic polynomial of a $N\times N$ complex Ginibre random matrix with the characteristic polynomial evaluated at a point in the unit disk. More precisely, we calculate the large $N$ asymptotics of $\mathbb{E}|\det(G_N-z)|^{\gamma}$, where $G_N$ is a $N\times N$ matrix whose entries are i.i.d and distributed as $N^{-1/2}Z$, $Z$ being a standard complex Gaussian, $\mathrm{Re}(\gamma)>-2$, and $|z|<1$. This expectation is proportional to the determinant of a complex moment matrix with a symbol which is supported in the whole complex plane and has a Fisher-Hartwig type of singularity: $\det(\int_\mathbb{C} w^{i}\overline{w}^j |w-z|^\gamma e^{-N|w|^{2}}d^2 w)_{i,j=0}^{N-1}$. We study the asymptotics of this determinant using recent results due to Lee and Yang concerning the asymptotics of orthogonal polynomials with respect to the weight $|w-z|^\gamma e^{-N|w|^2}d^2 w$ along with differential identities familiar from the study of asymptotics of Toeplitz and Hankel determinants with Fisher-Hartwig singularities. To our knowledge, even in the case of one singularity, the asymptotics of the determinant of such a moment matrix whose symbol has support in a two-dimensional set and a Fisher-Hartwig singularity, have been previously unknown.

Asymptotics for Hankel Determinants Associated to a Hermite Weight with a Varying Discontinuity

We study $n\times n$ Hankel determinants constructed with moments of a Hermite weight with a Fisher-Hartwig singularity on the real line. We consider the case when the singularity is in the bulk and is both of root-type and jump-type. We obtain large $n$ asymptotics for these Hankel determinants, and we observe a critical transition when the size of the jumps varies with $n$. These determinants arise in the thinning of the generalised Gaussian unitary ensembles and in the construction of special function solutions of the Painlev\'e IV equation.

Gaussian Unitary Ensemble with Boundary Spectrum Singularity and σ‐Form of the Painlevé II Equation

AbstractWe consider the Gaussian unitary ensemble perturbed by a Fisher–Hartwig singularity simultaneously of both root type and jump type. In the critical regime where the singularity approaches the soft edge, namely, the edge of the support of the equilibrium measure for the Gaussian weight, the asymptotics of the Hankel determinant and the recurrence coefficients, for the orthogonal polynomials associated with the perturbed Gaussian weight, are obtained and expressed in terms of a family of smooth solutions to the Painlevé XXXIV equation and the σ‐form of the Painlevé II equation. In addition, we further obtain the double scaling limit of the distribution of the largest eigenvalue in a thinning procedure of the conditioning Gaussian unitary ensemble, and the double scaling limit of the correlation kernel for the critical perturbed Gaussian unitary ensemble. The asymptotic properties of the Painlevé XXXIV functions and the σ‐form of the Painlevé II equation are also studied.

Emergence of a singularity for Toeplitz determinants and Painlevé V

We obtain asymptotic expansions for Toeplitz determinants corresponding to a family of symbols depending on a parameter $t$. For $t$ positive, the symbols are regular so that the determinants obey Szeg\H{o}'s strong limit theorem. If $t=0$, the symbol possesses a Fisher-Hartwig singularity. Letting $t\to 0$ we analyze the emergence of a Fisher-Hartwig singularity and a transition between the two different types of asymptotic behavior for Toeplitz determinants. This transition is described by a special Painlev\'e V transcendent. A particular case of our result complements the classical description of Wu, McCoy, Tracy, and Barouch of the behavior of a 2-spin correlation function for a large distance between spins in the two-dimensional Ising model as the phase transition occurs.

Superoptimal approximation for unbounded symbols

The superoptimal Frobenius approximation of Toeplitz matrices is considered in connection with the case of unbounded symbols. In particular, we use the superoptimal approximation as preconditioner for the CG method when a Fisher–Hartwig singularity is present in the symbol, with special regard to systems coming from times series and financial applications. A theoretical discussion concerning classical circulant preconditioners and a numerical comparison with the Strang and with the optimal approximations are presented particularly with reference to the presence of noise.

Convergence analysis of superoptimal PCG algorithm for Toeplitz systems with a Fisher–Hartwig singularity

Recently, Lu and Hurvich [Y. Lu, C. Hurvich, On the complexity of the preconditioned conjugate gradient algorithm for solving toeplitz systems with a Fisher–Hartwig singularity, SIAM J. Matrix Anal. Appl. 27 (2005) 638–653] used the preconditioned conjugate gradient method with the optimal circulant preconditioner proposed in Chan [T. Chan, An optimal circulant preconditioner for Toeplitz systems, SIAM J. Sci. Statist. Comput. 9 (1988) 766–771] for solving the Toeplitz system T n ( f ) x = b where the generating function f is given by f ( ω ) = | 1 - e - i ω | - 2 d h ( ω ) with d ∈ - 1 2 , 1 2 ⧹ { 0 } . The function h ( ω ) is positive continuous on [ - π , π ] and differentiable on [ - π , π ] ⧹ { 0 } . In this paper, we will use the superoptimal circulant preconditioner proposed by Tyrtyshnikov [E. Tyrtyshnikov, Optimal and superoptimal circulant preconditioners, SIAM J. Matrix Anal. Appl. 13 (1992) 459–473] to solve the same problem when 0 < d < 1 / 2 . Our convergence analysis shows that the number of iterations is bounded by O ( log 3 n ) and therefore the complexity of our algorithm is O ( n log 4 n ) . We notice that the numerical performance of superoptimal preconditioner is almost the same as that of the optimal preconditioner.

Unitarily Invariant Norms of Toeplitz Matrices with Fisher–Hartwig Singularities

We provide upper bounds for unitarily invariant norms of finite Toeplitz matrices generated by functions with a Fisher–Hartwig singularity. These bounds are sharp as the matrix size goes to infinity. Our results improve previous estimates and confirm a conjecture recently raised by Böttcher.

Schatten norms of Toeplitz matrices with Fisher-Hartwig singularities

The asymptotics of the Schatten norms of finite Toeplitz matrices generated by functions with a Fisher-Hartwig singularity are described as the matrix dimension n goes to infinity. The message of the paper is to reveal some kind of a kink: the pth Schatten norm increases as n to the power 1/p before the singularity reaches a critical point and as n to an exponent depending on the singularity beyond the critical point.

Two Elementary Derivations of the Pure Fisher-Hartwig Determinant

We present two elementary derivations of the formula for the Toeplitz determinant generated by a pure Fisher-Hartwig singularity.

Toeplitz Determinants with One Fisher–Hartwig Singularity

Letcbe a function defined on the unit circle with Fourier coefficients {cn}∞n=−∞. The Fisher–Hartwig conjecture describes the asymptotic behaviour of the determinants of then×nToeplitz matricesDn(c)=det[ci−j]n−1i, j=0for a certain class of functionsc. In this paper we prove this conjecture in the case of functions with one singularity. More precisely, we consider functions of the formc(eiθ)=b(eiθ)tβ(ei(θ−θ1))uα(ei(θ−θ1)).Heretβ(eiθ)=exp(iβ(θ−π)), 0<θ<2π, is a function with a jump discontinuity,uα(eiθ)=(2−2cosθ)αis a function which may have a zero, a pole, or a discontinuity of oscillating type, andbis a sufficiently smooth nonvanishing function with winding number equal to zero. The only restriction we impose on the parameters is that 2αis required not to be a negative integer. In the case where Reα⩽−1/2, i.e., where the corresponding functioncis not integrable, we identifycin an appropriate way with a distribution.

Toeplitz Operators and Determinants Generated by Symbols with One Fisher-Hartwig Singularity

Mathematische NachrichtenVolume 127, Issue 1 p. 95-123 Article Toeplitz Operators and Determinants Generated by Symbols with One Fisher-Hartwig Singularity Albrecht Böttcher, Albrecht Böttcher Technische Hochschule Sektion Mathematik DDR - 9010 Karl-Marx-Stadt PSF 964Search for more papers by this authorBernd Silbermann, Bernd Silbermann Technische Hochschule Sektion Mathematik DDR - 9010 Karl-Marx-Stadt PSF 964Search for more papers by this author Albrecht Böttcher, Albrecht Böttcher Technische Hochschule Sektion Mathematik DDR - 9010 Karl-Marx-Stadt PSF 964Search for more papers by this authorBernd Silbermann, Bernd Silbermann Technische Hochschule Sektion Mathematik DDR - 9010 Karl-Marx-Stadt PSF 964Search for more papers by this author First published: 1986 https://doi.org/10.1002/mana.19861270108Citations: 34AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Citing Literature Volume127, Issue11986Pages 95-123 RelatedInformation