Fredholm Determinant Research Articles

High-temperature analysis of the transverse dynamical two-point correlation function of the XX quantum-spin chain

We analyze the transverse dynamical two-point correlation function of the XX chain by means of a thermal form factor series. The series is rewritten in terms of the resolvent and the Fredholm determinant of an integrable integral operator. This connects it with a matrix Riemann-Hilbert problem. We express the correlation function in terms of the solution of the matrix Riemann-Hilbert problem. The matrix Riemann-Hilbert problem is then solved asymptotically in the high-temperature limit. This allows us to obtain the leading high-temperature contribution to the two-point correlation function at any fixed space-time separation.

Свойства алгебры псевдодифференциальных операторов, связанные с интегрируемыми иерархиями

In this paper we discuss and prove various properties of the algebra of pseudodifferential operators related to integrable hierarchies in this algebra, in particular the KPhierarchy and its strict version. Some explain the form of the equations involved or giveinsight in why certain equations in these systems are combined, others lead to additionalproperties of these systems like a characterization of the eigenfunctions of the linearizations ofthe mentioned hierarchies, the description of elementary Darboux transformations of bothhierarchies and the search for expressions in Fredholm determinants for the constructedeigenfunctions and their duals.

Sharp asymptotics for Fredholm Pfaffians related to interacting particle systems and random matrices

It has been known since the pioneering paper of Mark Kac, that the asymptotics of Fredholm determinants can be studied using probabilistic methods. We demonstrate the efficacy of Kac' approach by studying the Fredholm Pfaffian describing the statistics of both non-Hermitian random matrices and annihilating Brownian motions. Namely, we establish the following two results. Firstly, let $\sqrt{N}+\lambda_{max}$ be the largest real eigenvalue of a random $N\times N$ matrix with independent $N(0,1)$ entries (the `real Ginibre matrix'). Consider the limiting $N\rightarrow \infty$ distribution $\mathbb{P}[\lambda_{max}<-L]$ of the shifted maximal real eigenvalue $\lambda_{max}$. Then \[ \lim_{L\rightarrow \infty} e^{\frac{1}{2\sqrt{2\pi}}\zeta\left(\frac{3}{2}\right)L} \mathbb{P}\left(\lambda_{max}<-L\right) =e^{C_e}, \] where $\zeta$ is the Riemann zeta-function and \[ C_e=\frac{1}{2}\log 2+\frac{1}{4\pi}\sum_{n=1}^{\infty}\frac{1}{n} \left(-\pi+\sum_{m=1}^{n-1}\frac{1}{\sqrt{m(n-m)}}\right). \] Secondly, let $X_t^{(max)}$ be the position of the rightmost particle at time $t$ for a system of annihilating Brownian motions (ABM's) started from every point of $\mathbb{R}_{-}$. Then \[ \lim_{L\rightarrow \infty} e^{\frac{1}{2\sqrt{2\pi}}\zeta\left(\frac{3}{2}\right)L} \mathbb{P}\left(\frac{X_{t}^{(max)}}{\sqrt{4t}}<-L\right) =e^{C_e}. \] These statements are a sharp counterpart of our previous results improved by computing the terms of order $L^{0}$ in the asymptotic expansion of the corresponding Fredholm Pfaffian.

Differential Relations for the Solutions to the NLS Equation and Their Different Representations

Solutions to the focusing nonlinear Schr\"odinger equation (NLS) of order $N$ depending on $2N-2$ real parameters in terms of wronskians and Fredholm determinants are given. These solutions give families of quasi-rational solutions to the NLS equation denoted by $v_{N}$ and have been explicitly constructed until order $N = 13$. These solutions appear as deformations of the Peregrine breather $P_{N}$ as they can be obtained when all parameters are equal to $0$. These quasi rational solutions can be expressed as a quotient of two polynomials of degree $N(N+1)$ in the variables $x$ and $t$ and the maximum of the modulus of the Peregrine breather of order $N$ is equal to $2N+1$. \\ Here we give some relations between solutions to this equation. In particular, we present a connection between the modulus of these solutions and the denominator part of their rational expressions. Some relations between numerator and denominator of the Peregrine breather are presented.

Correlation functions of one-dimensional strongly interacting two-component gases

We address the problem of calculating the correlation functions of one-dimensional two-component gases with strong repulsive contact interactions. The model considered in this paper describes particles with fractional statistics and in appropriate limits reduces to the Gaudin-Yang model or the spinor Bose gas. In the case of impenetrable particles we derive a Fredholm determinant representation for the temperature-, time-, and space-dependent correlation functions which is very easy to implement numerically and constitute the starting point for the analytical investigation of the asymptotics. Making use of this determinant representation and the solution of an associated Riemann-Hilbert problem we derive the low-energy asymptotics of the correlators in the spin-incoherent regime characterized by near ground-state charge degrees of freedom but a highly thermally disordered spin sector. The asymptotics present features reminiscent of spin-charge separation with the spin part exponentially decaying in space separation and oscillating with a period proportional to the statistics parameter while the charge part presents scaling with anomalous exponents which cannot be described by any unitary conformal field theory. The momentum distribution and the Fourier transform of the dynamical Green's function are asymmetrical for arbitrary statistics, a direct consequence of the broken space-reversal symmetry. Due to the exponential decay the momentum distribution $n(k)$ at zero temperature does not present algebraic singularities but the tails obey the universal decay $\lim_{k\rightarrow\pm\infty}n(k)\sim C/k^4$ with the amplitude $C$ given by Tan's contact. As a function of the statistics parameter the contact is a monotonic function reaching its minimum for the fermionic system and the maximum for the bosonic system.

Spectral Features of the Solving of a Fredholm Homogeneous Integro-Differential Equation with Integral Conditions and Reflecting Deviation

The problems of solvability and construction of solutions of a nonlocal boundary value problem for the second-order Fredholm integro-differential equation with degenerate kernel, integral conditions, spectral parameters and reflecting deviation are considered. Using the method of the degenerate kernel, the boundary value problem is integrated as an ordinary differential equation. When we define the arbitrary integration constants there are possible five cases with respect to the first spectral parameter. Calculated values of the spectral parameter for each case. Further, the problem is reduced to solving systems of linear algebraic equations. Irregular values of the second spectral parameter are determined. At irregular values of the second spectral parameter the Fredholm determinant is degenerate. Other values of the second spectral parameter, for which the Fredholm determinant does not degenerate, are called regular values. Taking the values of the first spectral parameter into account for regular values of the second spectral parameter the corresponding solutions were constructed for each of five cases. The stability of the solution of the boundary value problem for given values in integral conditions is proved. The conditions under which the solution of the boundary value problem will be small are studied. For the irregular values of the second spectral parameter each of the five cases is checked separately. The orthogonality conditions are used. Cases are determined in which the problem has an infinite number of solutions and these solutions are constructed. For other cases, the absence of nontrivial solutions of the problem is proved.

Linear statistics of random matrix ensembles at the spectrum edge associated with the Airy kernel

In this paper, we study the large N behavior of the moment-generating function (MGF) of the linear statistics of N×N Hermitian matrices in the Gaussian unitary, symplectic, orthogonal ensembles (GUE, GSE, GOE) and Laguerre unitary, symplectic, orthogonal ensembles (LUE, LSE, LOE) at the edge of the spectrum. From the finite N Fredholm determinant expression of the MGF of the linear statistics [35], we find the large N asymptotics of the MGF associated with the Airy kernel in these Gaussian and Laguerre ensembles. Then we obtain the mean and variance of the suitably scaled linear statistics. We show that there is an equivalence between the large N behavior of the MGF of the scaled linear statistics in Gaussian and Laguerre ensembles, which leads to the statistical equivalence between the mean and variance of suitably scaled linear statistics in Gaussian and Laguerre ensembles. In the end, we use the Coulomb fluid method to obtain the mean and variance of another type of linear statistics in GUE, which reproduces the result of Basor and Widom [7].

Asymptotics of determinants of 4‐th order operators at zero

AbstractWe consider fourth order ordinary differential operators on the half‐line and on the line, where the perturbation has compactly supported coefficients. The Fredholm determinant for this operator is an analytic function in the whole complex plane without zero. We describe the determinant at zero. We show that in the generic case it has a pole of order 4 in the case of the line and of order 1 in the case of the half‐line.

The τ-function of the Ablowitz-Segur family of solutions to Painlevé II as a Widom constant

The τ-functions of certain Painlevé equations (PVI, PV, and PIII) can be expressed as Fredholm determinants. Furthermore, the minor expansion of these determinants provides an interesting connection to random partitions. This paper is a step toward understanding whether the τ-function of Painlevé II has a Fredholm determinant representation. The Ablowitz-Segur family of solutions are special one parameter solutions of Painlevé II, and the corresponding τ-function is known to be the Fredholm determinant of the Airy kernel. We develop a formalism for open contour in parallel to the one formulated in terms of a suitable combination of Toeplitz operators called the Widom constant and verify that the Widom constant for the Ablowitz-Segur family of solutions is indeed the determinant of the Airy kernel. Finally, we construct a suitable basis and obtain the minor expansion of the Ablowitz-Segur τ-function.

Computation of Coverage Probabilities in a Spherical Germ-Grain Model

We consider a spherical germ-grain model on $\mathbb {R}^{d}$ in which the centers of the spheres are driven by a possibly non-Poissonian point process. We show that various covering probabilities can be expressed using the cumulative distribution function of the random radii on one hand, and distances to certain subsets of $\mathbb {R}^{d}$ on the other hand. This result allows us to compute the spherical and linear contact distribution functions, and to derive expressions which are suitable for numerical computation. Determinantal point processes are an important class of examples for which the relevant quantities take the form of Fredholm determinants.

Borodin–Okounkov formula, string equation and topological solutions of Drinfeld–Sokolov hierarchies

We give a general method to compute the expansion of topological tau functions for Drinfeld-Sokolov hierarchies associated to an arbitrary untwisted affine Kac-Moody algebra. Our method consists of two main steps: first these tau functions are expressed as (formal) Fredholm determinants of the type appearing in the Borodin-Okounkov formula, then the kernels for these determinants are found using a reduced form of the string equation. A number of explicit examples are given.

Riemann–Hilbert approach to a generalized sine kernel

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.

Fredholm Determinant Solutions of the Painlevé II Hierarchy and Gap Probabilities of Determinantal Point Processes

AbstractWe study Fredholm determinants of a class of integral operators, whose kernels can be expressed as double contour integrals of a special type. Such Fredholm determinants appear in various random matrix and statistical physics models. We show that the logarithmic derivatives of the Fredholm determinants are directly related to solutions of the Painlevé II hierarchy. This confirms and generalizes a recent conjecture by Le Doussal, Majumdar, and Schehr [20]. In addition, we obtain asymptotics at $\pm \infty $ for the Painlevé transcendents and large gap asymptotics for the corresponding point processes.

Extreme eigenvalue distributions of Jacobi ensembles: New exact representations, asymptotics and finite size corrections

Let W1 and W2 be independent n×n complex central Wishart matrices with m1 and m2 degrees of freedom respectively. This paper is concerned with the extreme eigenvalue distributions of double-Wishart matrices (W1+W2)−1W1, which are analogous to those of F matrices W1W2−1 and those of the Jacobi unitary ensemble (JUE). Defining α1=m1−n and α2=m2−n with m1, m2≥n, we derive new exact distribution formulas in terms of (α1+α2)-dimensional matrix determinants, with entries involving derivatives of Legendre polynomials. This provides a convenient exact representation, while facilitating a direct large-n analysis with α1 and α2 fixed (i.e., under the so-called “hard-edge” scaling limit). The analysis is based on new asymptotic properties of Legendre polynomials and their relation with Bessel functions that are here established. Specifically, we present limiting formulas for the smallest and largest eigenvalue distributions as n→∞ in terms of α1- and α2-dimensional determinants respectively, which agrees with expectations from known universality results involving the JUE and the Laguerre unitary ensemble (LUE). We also derive finite-n corrections for the asymptotic extreme eigenvalue distributions under hard-edge scaling, giving new insights on universality by comparing with corresponding correction terms derived recently for the LUE. Our derivations are based on elementary algebraic manipulations and properties of Legendre polynomials, differing from existing results on double-Wishart and related models which often involve Fredholm determinants, Painlevé differential equations, or hypergeometric functions of matrix arguments.

Infinite-dimensional Log-Determinant divergences between positive definite Hilbert–Schmidt operators

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 resulting parametrized family of Alpha-Beta Log-Det divergences is general and contains many divergences between positive definite unitized Hilbert–Schmidt operators as special cases, including the infinite-dimensional affine-invariant Riemannian distance and the infinite-dimensional generalization of the symmetric Stein divergence.

Janossy densities for chiral random matrix ensembles and their applications to two-color QCD

We compute individual distributions of low-lying eigenvalues of massive chiral random matrix ensembles by the Nyström-type quadrature method for evaluating the Fredholm determinant and Pfaffian that represent the analytic continuation of the Janossy densities (conditional gap probabilities). A compact formula for individual eigenvalue distributions suited for precise numerical evaluation by the Nyström-type method is obtained in an explicit form, and the kth smallest eigenvalue distributions are numerically evaluated for chiral unitary and symplectic ensembles in the microscopic limit. As an application of our result, the low-lying Dirac spectra of the SU(2) lattice gauge theory with NF = 8 staggered flavors are fitted to the numerical prediction from the chiral symplectic ensemble, leading to a precise determination of the chiral condensate of a two-color QCD-like system in the future.

The Birman-Schwinger Operator for a Parabolic Quantum Well in a Zero-Thickness Layer in the Presence of a Two-Dimensional Attractive Gaussian Impurity

In this note we consider a quantum mechanical particle moving inside an infinitesimally thin layer constrained by a parabolic well in the $x$-direction and, moreover, in the presence of an impurity modelled by an attractive Gaussian potential. We investigate the Birman-Schwinger operator associated to a model assuming the presence of a Gaussian impurity inside the layer and prove that such an integral operator is Hilbert-Schmidt, which allows the use of the modified Fredholm determinant in order to compute the bound states created by the impurity. Furthermore, we consider the case where the Gaussian potential degenerates to a $\delta$-potential in the $x$-direction and a Gaussian potential in the $y$-direction. We construct the corresponding self-adjoint Hamiltonian and prove that it is the limit in the norm resolvent sense of a sequence of corresponding Hamiltonians with suitably scaled Gaussian potentials. Satisfactory bounds on the ground state energies of all Hamiltonians involved are exhibited.

Large Gap Asymptotics for Airy Kernel Determinants with Discontinuities

We obtain large gap asymptotics for Airy kernel Fredholm determinants with any number m of discontinuities. These m-point determinants are generating functions for the Airy point process and encode probabilistic information about eigenvalues near soft edges in random matrix ensembles. Our main result is that the m-point determinants can be expressed asymptotically as the product of m 1-point determinants, multiplied by an explicit constant pre-factor which can be interpreted in terms of the covariance of the counting function of the process.

The generating function for the Bessel point process and a system of coupled Painlevé V equations

We study the joint probability generating function for [Formula: see text] occupancy numbers on disjoint intervals in the Bessel point process. This generating function can be expressed as a Fredholm determinant. We obtain an expression for it in terms of a system of coupled Painlevé V equations, which are derived from a Lax pair of a Riemann–Hilbert problem. This generalizes a result of Tracy and Widom [C. A. Tracy and H. Widom, Level spacing distributions and the Bessel kernel, Commun. Math. Phys. 161(2) (1994) 289–309], which corresponds to the case [Formula: see text]. We also provide some examples and applications. In particular, several relevant quantities can be expressed in terms of the generating function, like the gap probability on a union of disjoint bounded intervals, the gap between the two smallest particles, and large [Formula: see text] asymptotics for [Formula: see text] Hankel determinants with a Laguerre weight possessing several jump discontinuities near the hard edge.

On the analyticity of the fredholm determinant

We prove that the Fredholm determinant is a real analytic function on the space of trace class operators defined on a separable Hilbert space H. This allows us to measure the size of sets of trace class and Schatten p-class operators which are not invertible and give an explicit description of their Aronszajn decomposition.