Relative Szegő Asymptotics for Toeplitz Determinants

2017 ◽  
Vol 2019 (17) ◽  
pp. 5441-5496
Author(s):  
Maurice Duits ◽  
Rostyslav Kozhan
Keyword(s):  
Limit Theorem ◽  
Wide Class ◽  
Toeplitz Determinants ◽  
Single Arc ◽  
Relative Version ◽  
Strong Szegö Limit Theorem ◽  
Szego Asymptotics ◽  
Essential Support ◽  
Verblunsky Coefficients ◽  
Second Order Asymptotics

Abstract We study the asymptotic behaviour, as $n \to \infty$, of ratios of Toeplitz determinants $D_n({\rm e}^h {\rm d}\mu)/D_n({\rm d}\mu)$ defined by a measure $\mu$ on the unit circle and a sufficiently smooth function $h$. The approach we follow is based on the theory of orthogonal polynomials. We prove that the second order asymptotics depends on $h$ and only a few Verblunsky coefficients associated to $\mu$. As a result, we establish a relative version of the Strong Szegő Limit Theorem for a wide class of measures $\mu$ with essential support on a single arc. In particular, this allows the measure to have a singular component within or outside of the arc.

Regenerative stochastic processes

1955 ◽  
Vol 232 (1188) ◽  
pp. 6-31 ◽  
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Probability Distribution ◽  
Stochastic Processes ◽  
Wide Class ◽  
Initial Conditions ◽  
Limiting Distribution ◽  
Regenerative Processes ◽  
Exponential Distributions ◽  
Negative Exponential

A wide class of stochastic processes, called regenerative, is defined, and it is shown that under general conditions the instantaneous probability distribution of such a process tends with time to a unique limiting distribution, whatever the initial conditions. The general results are then applied to 'S.M.-processes’, a generalization of Markov chains, and it is shown that the limiting distribution of the process may always be obtained by assuming negative exponential distributions for the ‘waits’ in the different ‘states’. Lastly, the behaviour of integrals of regenerative processes is considered and, amongst other results, an ergodic and a multi-dimensional central limit theorem are proved.

scholarly journals Matrix models, Toeplitz determinants and recurrence times for powers of random unitary matrices

2015 ◽  
Vol 04 (03) ◽  
pp. 1550011 ◽  
Author(s):  
O. Marchal
Keyword(s):  
Return Time ◽  
Unitary Matrix ◽  
Unitary Matrices ◽  
Toeplitz Determinants ◽  
Recurrence Times ◽  
Toeplitz Determinant ◽  
Large N ◽  
Single Arc ◽  
Random Unitary Matrices ◽  
First Return Time

The purpose of this paper is to study the eigenvalues [Formula: see text] of Ut where U is a large N×N random unitary matrix and t > 0. In particular we are interested in the typical times t for which all the eigenvalues are simultaneously close to 1 in different ways thus corresponding to recurrence times in the issue of quantum measurements. Our strategy consists in rewriting the problem as a random matrix integral and use loop equations techniques to compute the first-orders of the large N asymptotic. We also connect the problem to the computation of a large Toeplitz determinant whose symbol is the characteristic function of several arc segments of the unit circle. In particular in the case of a single arc segment we recover Widom's formula. Eventually we explain why the first return time is expected to converge toward an exponential distribution when N is large. Numerical simulations are provided along the paper to illustrate the results.

ASYMPTOTIC BEHAVIOR OF SOME HANKEL-TOEPLITZ DETERMINANTS

1992 ◽  
Vol 04 (01) ◽  
pp. 65-94 ◽  
Author(s):  
GEORGE A. BAKER ◽  
DANIEL BESSIS ◽  
PIERRE MOUSSA
Keyword(s):  
Asymptotic Behavior ◽  
Quantum Gravity ◽  
Quantum Groups ◽  
Wide Class ◽  
Continued Fraction ◽  
Finite Support ◽  
Toeplitz Determinants ◽  
Large Classes ◽  
Large Order ◽  
Recent Increase

Motivated by the recent increase in interest in Toeplitz determinants of large order, whose elements are moments with respect to measures, by their connection with the theory of quantum gravity we have given exact values of the determinants for several large classes of measures. These classes are related to the continued fraction of Gauss and its q-extension. We have proven that the q-extension, which is related to the theory of quantum groups, is unique. In addition we have extended the work of Szegő from measures of finite support to a wide class of those with infinite support.

scholarly journals Crossing bridges with strong Szegő limit theorem

2021 ◽  
Vol 2021 (4) ◽  
Author(s):  
A. V. Belitsky ◽  
G. P. Korchemsky
Keyword(s):  
Asymptotic Behavior ◽  
Limit Theorem ◽  
Strong Coupling ◽  
Fredholm Determinant ◽  
Form Factors ◽  
Strong Coupling Expansion ◽  
Bessel Operator ◽  
Point Correlation ◽  
A New Technique ◽  
Strong Szegö Limit Theorem

Abstract We develop a new technique for computing a class of four-point correlation functions of heavy half-BPS operators in planar $$ \mathcal{N} $$ 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.

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