Harold Widom
In previous work of C. A. Tracy and the author asymptotic formulas were derived for certain operator determinants whose interest lay in the fact that quotients of them gave solutions to the cylindrical Toda equations. In the present paper we consider a more general class of operators which retain some of the properties of those cited and we find analogous asymptotics for the determinants
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An asymptotic formula is found for a Toeplitz determinant with the symbol supported on an arc of the unit circle in the case when the symbol has Fisher-Hartwig singularities.
In this paper we determine the asymptotics of the determinants of truncated Wiener-Hopf plus Hankel operators $\det(W_R(a)\pm H_R(a))$ as $R$ tends to infinity for symbols $a(x)=(x^2/(1+x^2))^β$ with the parameter $β$ being of small size.
We prove that the asymptotics of the Fredholm determinant of $I-K_α$, where $K_α$ is the integral operator with the sine kernel $\sin(x-y)/(x-y)/π$ on the interval $[0,α]$ is given by a formula which was conjectured by F.J. Dyson. The first and second order asymptotics as well as the higher order asymptotics except for the constant term have already been proved. In this paper we thus determine the constant term.
We compute the asymptotics of the determinants of certain $n\times n$ Toeplitz + Hankel matrices $T_n(a)+H_n(b)$ as $n\to\infty$ with symbols of Fisher-Hartwig type. More specifically we consider the case where $a$ has zeros and poles and where $b$ is related to $a$ in specific ways. Previous results of Deift, Its and Krasovsky dealt with the case where $a$ is even. We are generalizing this in a mild way to certain non-even symbols.
In this paper we are going to prove two asymptotic formulas for determinants det(I-K_s), as s goes to infinity, where K_s are the Wiener-Hopf-Hankel operators acting on L^2[0,s] with the kernels K(x-y)+K(x+y) and K(x-y)-K(x+y), respectively, and K(t):=sin(t)/(π*t). These formulas were conjectured by Dyson. The identification of the constant term in the asymptotics was an open problem for a long time.
In this paper we establish several relations between the determinants of the following structured matrices: Hankel matrices, symmetric Toeplitz + Hankel matrices and Toeplitz matrices. Using known results for the asymptotic behavior of Toeplitz determinants, these identities are used in order to obtain Fisher-Hartwig type results on the asymptotics of certain skewsymmetric Toeplitz determinants and certain Hankel determinants.
We consider a perturbation determinant for pairs of nonpositive (in a sense of Komatsu) operators on Banach space with nuclear difference and prove a generalization of the important formula for the logarithmic derivative of this determinant. To this end the Frechet differentiability of operator monotonic (negative complete Bernstein) functions of negative and nonpositive operators on Banach spaces is investigated. The results may be regarded as a contribution to the Hirsch functional calculus.
Integrable integral operator can be studied by means of a matrix Riemann--Hilbert problem. However, in the case of so-called integrable operators with shifts, the associated Riemann--Hilbert problem becomes operator valued and this complicates strongly the analysis. In this note, we show how to circumvent, in a very simple way, the use of such a setting while still being able to characterize the large-$x$ asymptotic behavior of the determinant associated with the operator.
We establish an asymptotic formula for determinants of truncated Wiener-Hopf+Hankel operators with symbol equal to the exponential of a constant times the characteristic function of an interval. This is done by reducing it to the corresponding (known) asymptotics for truncated Toeplitz+Hankel operators. The determinants in question arise in random matrix theory in determining the limiting distribution for the number of eigenvalues in an interval for a scaled Laguerre ensemble of positive Hermitian matrices.
The current work generalizes the author's previous work on the infinite-dimensional Alpha Log-Determinant (Log-Det) divergences and Alpha-Beta Log-Det divergences, defined on the set of positive definite unitized trace class operators on a Hilbert space, to the entire Hilbert manifold of positive definite unitized Hilbert-Schmidt operators. This generalization is carried out via the introduction of the extended Hilbert-Carleman determinant for unitized Hilbert-Schmidt operators, in addition to the previously introduced extended Fredholm determinant for unitized trace class operators. The resul…
A survey of asymptotics of determinants for structured matrices
本综述介绍如何利用算子理论方法生成结构化矩阵行列式的渐近性，涵盖有限 Toeplitz 矩阵、Toeplitz 加 Hankel 矩阵及其推广，并首次提出部分新结果。
We extend the classical trace (and determinant) known for the integral operators $$ ({\mathcal I}+)\int_{[0,1)^N}{\bf A}({\bf k},{\bf x}){\bf u}({\bf x})d{\bf x} $$ with matrix-valued kernels ${\bf A}$ to the operators of the form $$ \sum_α\int_{[0,1)^{|α|}}{\bf A}({\bf k},{\bf x}_α){\bf u}({\bf k}_{\overlineα},{\bf x}_α)d{\bf x}_α, $$ where $α$ are arbitrary subsets of the set $\{1,...,N\}$. Such operators form a Banach algebra containing simultaneously all integral operators of the dimensions $\leqslant N$. In this sense, it is a largest algebra where explicit traces and determinants are con…
Trace, determinant and eigenvalues of nuclear operators
该研究将行列式、迹理论与拟赋范张量积理论的结果结合，推导巴拿赫空间中核算子特征值分布及谱迹与核算迹一致性的新定理，并以Lorentz–Lapresté广义核算子类为例进行分析。
The solution of a problem arising in integrable systems requires sharp asymptotics for the inverses and determinants of truncated Wiener-Hopf operators, both in the regular case (where the non-truncated Wiener-Hopf operator is invertible) and in singular cases. This paper treats the singular cases where the symbol of the Wiener-Hopf operator has one double zero or two simple zeros. We find formulas for the inverses that hold uniformly throughout the underlying interval with very small error, and formulas for the determinants with very small error.
By using Riemann--Hilbert problem based techniques, we obtain the asymptotic expansion of lacunary Toeplitz determinants $\det_N\big[ c_{\ell_a-m_b}[f] \big]$ generated by holomorhpic symbols, where $\ell_a=a$ (resp. $m_b=b$) except for a finite subset of indices $a=h_1,\dots, h_n$ (resp. $b=t_1,\dots, t_r$). In addition to the usual Szegö asymptotics, our answer involves a determinant of size $n+r$.
We prove asymptotic formulas for Toeplitz determinants generated by functions with sequences of Fourier coefficients belonging to weighted Orlicz sequence classes. We concentrate our attention on the case of nonvanishing generating functions with nonzero Cauchy index.
In this article we derive, using standard methods of Toeplitz theory, an asymptotic formula for certain large minors of Toeplitz matrices. D. Bump and P. Diaconis obtained the same asymptotics using representation theory, with an answer having a different form.
We establish asymptotic formulas for the determinants of finite Toeplitz + Hankel matrices of size N, as N goes to infinity for singular generating functions defined on the unit circle in the special case where the generating function is even, i.e., where the Toeplitz + Hankel matrices are symmetric.
In work of C. A. Tracy and the author asymptotic formulas were derived for certain operator determinants which gave solutions to the cylindrical Toda equations. Later the author considered a more general class of operators which retained some of the properties of those cited and found analogous asymptotics for the determinants. These results applied only in what we call the regular case, when the symbol of an associated convolution operator is nonzero and has zero index. The present work goes further and establishes asymptotics for a class of singular cases.
The authors use Riemann-Hilbert methods to compute the constant that arises in the asymptotic behavior of the Airy-kernel determinant of random matrix theory.
The paper is devoted to exact and asymptotic formulas for the determinants of Toeplitz matrices with perturbations by blocks of fixed size in the four corners. If the norms of the inverses of the unperturbed matrices remain bounded as the matrix dimension goes to infinity, then standard perturbation theory yields asymptotic expressions for the perturbed determinants. This premise is not satisfied for matrices generated by so-called Fisher-Hartwig symbols. In that case we establish formulas for pure single Fisher-Hartwig singularities and for Hermitian matrices induced by general Fisher-Hartwig…
In this expository article we describe the asymptotics of certain Fredholm determinants which provide solutions to the cylindrical Toda equations, and we explain how these asymptotics are derived. The connection with Fredholm determinants arising in the theory of random matrices, and their asymptotics, are also discussed.
We obtain "large gap" asymptotics for a Fredholm determinant with a confluent hypergeometric kernel. We also obtain asymptotics for determinants with two types of Bessel kernels which appeared in random matrix theory.
The purpose of this paper is to describe asymptotic formulas for determinants of a sum of finite Toeplitz and Hankel matrices with singular generating functions. The formulas are similar to those of the analogous problem for finite Toeplitz matrices for a certain class of symbols. However, the appearance of the Hankel matrices changes the nature of the asymptotics in some instances depending on the location of the singularities. Several concrete examples are also described in the paper.
In this paper we determine the asymptotics of the determinant of Bessel operators for sufficiently smooth generating functions. These operators are similar to Wiener-Hopf operators with the Fourier transform replaced by the Hankel transform and thus the asymptotics of the determinanst are similar to the well-known Szegö-Akhiezer-Kac formula for truncated Wiener-Hopf determinants. In order to compute the above, we also show that the Bessel operators differ from the Wiener-Hopf by a Hilbert-Schmidt operator.
In Random Matrix Theory the local correlations of the Laguerre and Jacobi Unitary Ensemble in the hard edge scaling limit can be described in terms of the Bessel kernel (containing a parameter $α$). In particular, the so-called hard edge gap probabilities can be expressed as the Fredholm determinants of the corresponding integral operator restricted to the finite interval [0, R]. Using operator theoretic methods we are going to compute their asymptotics as R goes to infinity under certain assumption on the parameter $α$.
We compute the asymptotics of a block Toeplitz determinant which arises in the classical dimer model for the triangular lattice when considering the monomer-monomer correlation function. The model depends on a parameter interpolating between the square lattice ($t=0$) and the triangular lattice ($t=1$), and we obtain the asymptotics for $0<t\le 1$. For $0<t<1$ we apply the Szegö Limit Theorem for block Toeplitz determinants. The main difficulty is to evaluate the constant term in the asymptotics, which is generally given only in a rather abstract form.
A determinant approach for generalized $q$-Bernoulli polynomials and asymptotic results
本文针对生成函数含 Jackson q-类似 Bessel 函数的三族广义 q-伯努利多项式，给出其行列式表示、大 n 渐近结果及整函数展开定理，并举例说明。
We study the asymptotics in n for n-dimensional Toeplitz determinants whose symbols possess Fisher-Hartwig singularities on a smooth background. We prove the general non-degenerate asymptotic behavior as conjectured by Basor and Tracy. We also obtain asymptotics of Hankel determinants on a finite interval as well as determinants of Toeplitz+Hankel type. Our analysis is based on a study of the related system of orthogonal polynomials on the unit circle using the Riemann-Hilbert approach.
Cholesky decompositions of integral operators and the Fredholm determinant
本文将矩阵Cholesky分解的行列式公式推广到积分算子情形，提出并证明了$L^2[0,1]$上正定积分算子Fredholm行列式的等价计算公式。
We provide an alternative proof of the classical single-term asymptotics for Toeplitz determinants whose symbols possess Fisher-Hartwig singularities. We also relax the smoothness conditions on the regular part of the symbols and obtain an estimate for the error term in the asymptotics. Our proof is based on the Riemann-Hilbert analysis of the related systems of orthogonal polynomials and on differential identities for Toeplitz determinants. The result discussed in this paper is crucial for the proof of the asymptotics in the general case of Fisher-Hartwig singularities and extensions to Hanke…
Let $A = (a_{j,k})_{j,k=-\infty}^\infty$ be a bounded linear operator on $l^2(\mathbb{Z})$ whose diagonals $D_n(A) = (a_{j,j-n})_{j=-\infty}^\infty\in l^\infty(\mathbb{Z})$ are almost periodic sequences. For certain classes of such operators and under certain conditions, we are going to determine the asymptotics of the determinants $\det A_{n_1,n_2}$ of the finite sections of the operator $A$ as their size $n_2 - n_1$ tends to infinity. Examples of such operators include block Toeplitz operators and the almost Mathieu operator.
Determinants of elliptic boundary problems for Dirac operators I. Local boundary conditions
本文研究了有边流形上Dirac算子的泛函行列式，并利用Calderón投影讨论了边值问题的椭圆性。对于局部边界条件，我们给出了一个将这些行列式与相应的格林函数联系起来的明确公式。最后，我们将这一结果的情况下，一个二维盘袋状条件。
A determinant formula for Toeplitz operators associated to a minimal flow
针对极小流对应的 Toeplitz 代数，本文定义行列式并给出基于符号的公式，同时揭示其在底空间函数代数 K-理论中的应用。
We study the trace class perturbations of the half-line, discrete Laplacian and obtain a new bound for the perturbation determinant of the corresponding non-self-adjoint Jacobi operator. Based on this bound, we obtain the Lieb--Thirring inequalities for such operators. The spectral enclosure for the discrete spectrum and embedded eigenvalues are also discussed.
With localization techniques one can obtain general limit theorems for Toeplitz determinants with Fisher-Hartwig singularities from the asymptotics for any symbol with one singularity of general type. There exists a family of these for which the determinants can be evaluated explicitly and their asymptotics determined. But for the Wiener-Hopf analogue, although there are likely analogous localization techniques, there is not a single example known of a symbol with Fisher-Hartwig singularity for which the determinant can be evaluated explicitly. In this paper we determine the asymptotics of Wie…
We investigate the short-time expansion of the heat kernel of a Laplace type operator on a compact Riemannian manifold and show that the lowest order term of this expansion is given by the Fredholm determinant of the Hessian of the energy functional on a space of finite energy paths. This is the asymptotic behavior to be expected from formally expressing the heat kernel as a path integral and then (again formally) using Laplace's method on the integral. We also relate this to the zeta determinant of the Jacobi operator, which is another way to assign a determinant to the Hessian of the energy …
We study a class of rotation invariant determinantal ensembles in the complex plane; examples include the eigenvalues of Gaussian random matrices and the roots of certain families of random polynomials. The main result is a criteria for a central limit theorem to hold for angular statistics of the points. The proof exploits an exact formula relating the generating function of such statistics to the determinant of a perturbed Toeplitz matrix.
The purpose of this paper is to describe asymptotic formulas for determinants of certain operators that are analogues of Wiener-Hopf operators. The determinant formulas yield information about the distribution functions for certain random variables that arise in random matrix theory when one rescales at the edge of the spectrum.
Consider an h-pseudodifferential operator P, whose symbol extends holomorphically to a tubular neighborhood of the real phase space and converges sufficiently fast to 1, so that the determinant of P is well-defined. We show that the modulus of this determinant is asymptotically bounded by an exponential of the integral of the logarithm of the modulus of the symbol along a certain complex deformation of the real phase space. Since there are many possible such deformations, we get a variational problem. The paper is devoted to the corresponding variational calculus.
We study functional determinants for Dirac operators on manifolds with boundary. We give, for local boundary conditions, an explicit formula relating these determinants to the corresponding Green functions. We finally apply this result to the case of a bidimensional disk under bag-like conditions.
We give a survey on generalized Krein algebras $K_{p,q}^{α,β}$ and their applications to Toeplitz determinants. Our methods originated in a paper by Mark Krein of 1966, where he showed that $K_{2,2}^{1/2,1/2}$ is a Banach algebra. Subsequently, Widom proved the strong Szegő limit theorem for block Toeplitz determinants with symbols in $(K_{2,2}^{1/2,1/2})_{N\times N}$ and later two of the authors studied symbols in the generalized Krein algebras $(K_{p,q}^{α,β})_{N\times N}$, where $λ:=1/p+1/q=α+β$ and $λ=1$. We here extend these results to $0<λ<1$. The entire paper is based on fundamental wor…
A formula relating quotients of determinants of elliptic differential operators sharing their principal symbol, with local boundary conditions, to the corresponding Green function is given.
Transfer operators M_k acting on k-forms in R^n are associated to smooth transversal local diffeomorphisms and compactly supported weight functions. A formal trace is defined by summing the product of the weight and the Lefschetz sign over all fixed points of all the diffeos. This yields a formal Ruelle-Lefschetz determinant Det^#(1-zM). We use the Milnor-Ruelle-Kitaev equality (recently proved by Baillif), which expressed Det^#(1-zM) as an alternated product of determinants of kneading operators,Det(1+D_k(z)), to relate zeroes and poles of the Ruelle-Lefschetz determinant to the spectra of th…
We obtain an asymptotic formula for $n\times n$ Toeplitz determinants as $n\to \infty$, for real valued symbols with any fixed number of Fisher-Hartwig singularities, which is uniform with respect to the location of the singularities. As an application, we prove a conjecture by Fyodorov and Keating regarding moments of averages of the characteristic polynomial of the Circular Unitary Ensemble. In addition, we obtain an asymptotic formula regarding the momentum of impenetrable bosons in one dimension with periodic boundary conditions.
We study extremal properties of the determinant of Friederichs selfadjoint Laplacian on the Euclidean isosceles triangle envelopes of fixed area as a function of angles. Small-angle asymptotics show that the determinant grows without any bound as an angle of triangle envelope goes to zero. We prove that the equilateral triangle envelope (the most symmetrical geometry) always gives rise to a critical point of the determinant and find the critical value. Moreover, if the area of envelopes is not too large, then the determinant achieves its absolute minimum only on the equilateral triangle envelo…
We introduce a generalized trace functional TR in the spirit of Kontsevich and Vishik's canonical trace for classical SG-pseudodifferential operators on R^n and suitable manifolds, using a finite-part integral regularization technique. This allows us to define a zeta-regularized determinant det A for classical parameter-elliptic SG-operators A of order (μ,m), with μ>0, m\ge0. For m=0, the asymptotics of TR exp(-tA) as t\to 0 and of TR (λ-A)^{-k}$ as |λ|\to\infty are derived. For suitable pairs (A,A_0) we show that det A/det A_0 coincides with the so-called relative determinant det(A,A_0).
Suppose that $ϕ$ and $ψ$ are smooth complex-valued functions on the circle that are invertible, have winding number zero with respect to the origin, and have meromorphic extensions to an open neighborhood of the closed unit disk. Let $T_ϕ$ and $T_ψ$ denote the Toeplitz operators with symbols $ϕ$ and $ψ$ respectively. We give an explicit formula for the determinant of $T_ϕT_ψT_ϕ^{-1} T_ψ^{-1}$ in terms of the products of the tame symbols of $ϕ$ and $ψ$ on the open unit disk.
We review the asymptotic behavior of a class of Toeplitz (as well as related Hankel and Toeplitz + Hankel) determinants which arise in integrable models and other contexts. We discuss Szego, Fisher-Hartwig asymptotics, and how a transition between them is related to the Painleve V equation. Certain Toeplitz and Hankel determinants reduce, in certain double-scaling limits, to Fredholm determinants which appear in the theory of group representations, in random matrices, random permutations and partitions. The connection to Toeplitz determinants helps to evaluate the asymptotics of related Fredho…
We establish Plemelj-Smithies formulas for determinants in different algebras of operators. In particular we define a Poincaré type determinant for operators on the torus $\Tn$ and deduce formulas for determinants of periodic pseudo-differential operators in terms of the symbol. On the other hand, by applying a recently introduced notion of invariant operators relative to fixed decompositions of Hilbert spaces we also obtain formulae for determinants with respect to the trace class. The analysis makes use of the corresponding notion of full matrix-symbol. We also derive explicit formulas for d…
Speed of convergence in the Central Limit Theorem for the determinantal point process with the Bessel kernel
本文研究由汉克尔变换对角化的线性算子的弗雷德霍姆行列式渐近展开，推导出渐近小因子的显式公式，并将其应用于贝塞尔核行列式点过程，计算了加性泛函在柯尔莫哥洛夫-斯米尔诺夫度量下的中心极限定理收敛速度。
Let $L$ be a self-adjoint invertible operator in a Hilbert space such that $L^{-1}$ is $p$-summable. Under a certain discrete dimension spectrum assumption on $L$, we study the relation between the (regularized) Fredholm determinant, $\det_{p}(I+z\cdot L^{-1})$, on the one hand and the zeta regularized determinant, $\det_ζ(L+z)$, on the other. One of the main results is the formula \begin{equation*} \frac{\det_ζ (L + z)}{\det_ζ (L)} = \exp \left( \sum_{j=1}^{p-1} \frac{z^j}{j!} \cdot \frac{d^j}{dz^j} \log \det\nolimits_ζ (L+z) |_{z=0} \right) \cdot \det\nolimits_{p}(I + z \cdot L^{-1} ). \end{…
We consider the probability of two large gaps (intervals without eigenvalues) in the bulk scaling limit of the Gaussian Unitary Ensemble of random matrices. We determine the multiplicative constant in the asymptotics. We also provide the full explicit asymptotics (up to decreasing terms) for the transition between one and two large gaps.
We consider the quantum effective action of Dirac fermions on four dimensional flat Euclidean space coupled to external vector- and axial Yang-Mills fields, i.e., the logarithm of the (regularized) determinant of a Dirac operator on flat R^4 twisted by generalized Yang-Mills fields. According to physics folklore, the logarithmic divergent part of this effective action in the pure vector case is proportional to the Yang-Mills action. We present an explicit computation proving this fact, generalized to the chiral case. We use an efficient computation method for quantum effective actions which is…
We obtain asymptotics for Toeplitz, Hankel, and Toeplitz+Hankel determinants whose symbols possess Fisher-Hartwig singularities. Details of the proofs will be presented in another publication.
Numerical Fredholm determinants for matrix-valued kernels on the real line
本研究拓展了Bornemann的分析，提出一种计算实线上矩阵值核定义的迹类、希尔伯特-施密特积分算子Fredholm行列式的数值方法，推导了截断与求积误差估计，并在Birman-Schwinger算子上验证了效果。
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