Sine Kernel Research Articles

On some properties of generalized cardinal sine kernel fractional operators: Advantages and applications of the extended operators

A new Caputo-type fractional derivative model with a generalized cardinal sine kernel and its singular kernel extension were presented. In this paper, we used the Laplace transform as an effective tool to study the considered fractional derivative models. Then, we introduced the Riemann–Liouville-type for the studied generalized cardinal sine fractional derivative model. Next, a general framework of the generalized cardinal sine kernel fractional derivative model and its singular kernel extension in relation to functions is presented. Some properties of the studied fractional derivative operators and relationships such as the relation between fractional integral and derivative operators are discussed. The dynamics of some nonlinear fractional order models are simulated, using a numerical algorithm formulated in this paper, to demonstrate the motivations of using the extended operators. The extended version of the considered fractional derivative operators provided useful suggestions regarding the modeling issue.

Solving Saint Venant torsion problems for rectangular beams using single finite Fourier sine transform method

This research presents the single Fourier sine transform method (SFSTM) for solving the Saint Venant torsion problem of rectangular prismatic bars. The problem is a common theme in the theory of elasticity of unrestrained torsion which was previously expressed by Prandtl using Prandtl stress functions ϕ(x,y) as a Poisson type nonhomogeneous partial differential equation (PDE) called the stress compatibility equation. In this work the SFSTM was applied to the stress compatibility equation, converting the PDE to an easier to solve ordinary differential equation (ODE) in the transformed domain. The boundary conditions were used to find the integration constant and inversion was used to find the solution in the physical domain. The non vanishing stresses and torsional moments were thus found as a single series of infinite terms with rapid convergence. The maximum stresses and moments were found in standard form in terms of torsional parameters which were tabulated for various ratios of the cross-sectional dimensions. A comparison of the torsional parameters with previous results show that the present results are identical with previous results illustrating the accuracy of the SFSTM used. The sine kernel of the SFSTM satisfies the boundary conditions of the problem and contributed to the exact solution obtained. The SFSTM simplified the PDE to an ODE which is simpler to solve.

Asymptotics of noncolliding q-exchangeable random walks

We consider a process of noncolliding q-exchangeable random walks on making steps 0 (‘straight’) and −1 (‘down’). A single random walk is called q-exchangeable if under an elementary transposition of the neighboring steps the probability of the trajectory is multiplied by a parameter . Our process of m noncolliding q-exchangeable random walks is obtained from the independent q-exchangeable walks via the Doob’s h-transform for a nonnegative eigenfunction h (expressed via the q-Vandermonde product) with the eigenvalue less than 1. The system of m walks evolves in the presence of an absorbing wall at 0. The repulsion mechanism is the q-analogue of the Coulomb repulsion of random matrix eigenvalues undergoing Dyson Brownian motion. However, in our model, the particles are confined to the positive half-line and do not spread as Brownian motions or simple random walks. We show that the trajectory of the noncolliding q-exchangeable walks started from an arbitrary initial configuration forms a determinantal point process, and express its kernel in a double contour integral form. This kernel is obtained as a limit from the correlation kernel of q-distributed random lozenge tilings of sawtooth polygons. In the limit as , with γ > 0 fixed, and under a suitable scaling of the initial data, we obtain a limit shape of our noncolliding walks and also show that their local statistics are governed by the incomplete beta kernel. The latter is a distinguished translation invariant ergodic extension of the two-dimensional discrete sine kernel.

A new fractional derivative operator with generalized cardinal sine kernel: Numerical simulation

In this paper, we proposed a new fractional derivative operator in which the generalized cardinal sine function is used as a non-singular analytic kernel. In addition, we provided the corresponding fractional integral operator. We expressed the new fractional derivative and integral operators as sums in terms of the Riemann–Liouville fractional integral operator. Next, we introduced an efficient extension of the new fractional operator that includes integrable singular kernel to overcome the initialization problem for related differential equations. We also proposed a numerical approach for the numerical simulation of IVPs incorporating the proposed extended fractional derivatives. The proposed fractional operators, the developed relations and the presented numerical method are expected to be employed in the field of fractional calculus.

Entanglement entropies of an interval in the free Schrödinger field theory at finite density

We study the entanglement entropies of an interval on the infinite line in the free fermionic spinless Schrödinger field theory at finite density and zero temperature, which is a non-relativistic model with Lifshitz exponent z = 2. We prove that the entanglement entropies are finite functions of one dimensionless parameter proportional to the area of a rectangular region in the phase space determined by the Fermi momentum and the length of the interval. The entanglement entropy is a monotonically increasing function. By employing the properties of the prolate spheroidal wave functions of order zero or the asymptotic expansions of the tau function of the sine kernel, we find analytic expressions for the expansions of the entanglement entropies in the asymptotic regimes of small and large area of the rectangular region in the phase space. These expansions lead to prove that the analogue of the relativistic entropic C function is not monotonous. Extending our analyses to a class of free fermionic Lifshitz models labelled by their integer dynamical exponent z, we find that the parity of this exponent determines the properties of the bipartite entanglement for an interval on the line.

Critical edge behavior in the singularly perturbed Pollaczek–Jacobi type unitary ensemble

We investigate the orthogonal polynomials associated with a singularly perturbed Pollaczek–Jacobi type weight [Formula: see text] where [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text]. Based on our observation, we find that this weight includes the symmetric constraint [Formula: see text]. Our main results obtained here include two aspects: (1) Strong asymptotics: we deduce strong asymptotics of monic orthogonal polynomials with respect to the above weight function in different regions in the complex plane when the polynomial degree [Formula: see text] goes to infinity. Because of the effect of [Formula: see text] for varying [Formula: see text], the asymptotic behavior in a neighborhood of point [Formula: see text] is described in terms of the Airy function as [Formula: see text], but the Bessel function as [Formula: see text]. Due to symmetry, the similar local asymptotic behavior near the singular point [Formula: see text] can be derived. (2) Limiting eigenvalue correlation kernels: We calculate the limit of the eigenvalue correlation kernel of the corresponding unitary random matrix ensemble in the bulk of the spectrum described by the sine kernel, and at both sides of hard edge, expressed as a Painlevé III kernel. Our analysis is based on the Deift–Zhou nonlinear steepest descent method for Riemann–Hilbert problems.

Kernels for non interacting fermions via a Green’s function approach with applications to step potentials

The quantum correlations of N non interacting spinless fermions in their ground state can be expressed in terms of a two-point function called the kernel. Here we develop a general and compact method for computing the kernel in a general trapping potential in terms of the Green’s function for the corresponding single particle Schrödinger equation. For smooth potentials in one dimension the method allows a simple alternative derivation of the local density approximation for the density and of the sine kernel in the bulk part of the trap in the large N limit. It also recovers the density and the kernel of the so-called Airy gas at the edge. This method allows the analysis of the quantum correlations in the ground state when the potential has a singular part with a fast variation in space. For the square step barrier of height V 0 in one dimension, we derive explicit expressions for the density and for the kernel. For large Fermi energy μ > V 0 it describes the interpolation between two regions of different densities in a Fermi gas, each described by a different sine kernel. Of particular interest is the critical point of the square well potential when μ = V 0. In this critical case, while there is a macroscopic number of fermions in the lower part of the step potential, there is only a finite O(1) number of fermions on the shoulder, and moreover this number is independent of μ. In particular, the density exhibits an algebraic decay ∼1/x 2, where x is the distance from the jump. Furthermore, we show that the critical behavior around μ = V 0 exhibits universality with respect to the shape of the barrier. This is established (i) by an exact solution for a smooth barrier (the Woods–Saxon potential) and (ii) by establishing a general relation between the large distance behavior of the kernel and the scattering amplitudes of the single-particle wave-function.

Mesoscopic eigenvalue density correlations of Wigner matrices

We investigate to what extent the microscopic Wigner-Gaudin-Mehta-Dyson (WGMD) (or sine kernel) statistics of random matrix theory remain valid on mesoscopic scales. To that end, we compute the connected two-point spectral correlation function of a Wigner matrix at two mesoscopically separated points. In the mesoscopic regime, density correlations are much weaker than in the microscopic regime. Our result is an explicit formula for the two-point function. This formula implies that the WGMD statistics are valid to leading order on all mesoscopic scales, that in the real symmetric case there are subleading corrections matching precisely the WGMD statistics, while in the complex Hermitian case these subleading corrections are absent. We also uncover non-universal subleading correlations, which dominate over the universal ones beyond a certain intermediate mesoscopic scale. The proof is based on a hierarchy of Schwinger-Dyson equations for a sufficiently large class of polynomials in the entries of the Green function. The hierarchy is indexed by a tree, whose depth is controlled using stopping rules. A key ingredient in the derivation of the stopping rules is a new estimate on the density of states, which we prove to have bounded derivatives of all order on all mesoscopic scales.

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.

Boundaries of sine kernel universality for Gaussian perturbations of Hermitian matrices

We explore the boundaries of sine kernel universality for the eigenvalues of Gaussian perturbations of large deterministic Hermitian matrices. Equivalently, we study for deterministic initial data the time after which Dyson’s Brownian motion exhibits sine kernel correlations. We explicitly describe this time span in terms of the limiting density and rigidity of the initial points. Our main focus lies on cases where the initial density vanishes at an interior point of the support. We show that the time to reach universality becomes larger if the density vanishes faster or if the initial points show less rigidity.

Free fermions and α-determinantal processes

The -determinant is a one-parameter generalisation of the standard determinant, with corresponding to the determinant, and corresponding to the permanent. In this paper a simple limit procedure to construct -determinantal point processes out of fermionic processes is examined. The procedure is illustrated for a model of N free fermions in a harmonic potential. When the system is in the ground state, the rescaled correlation functions converge for large N to determinants (of the sine kernel in the bulk and the Airy kernel at the edges). We analyse the point processes associated to a special family of excited states of fermions and show that appropriate scaling limits generate -determinantal processes. Links with wave optics and other random matrix models are suggested.

Universality for conditional measures of the sine point process

The sine process is a rigid point process on the real line, which means that for almost all configurations X, the number of points in an interval I=[−R,R] is determined by the points of X outside of I. In addition, the points in I are an orthogonal polynomial ensemble on I with a weight function that is determined by the points in X∖I. We prove a universality result that in particular implies that the correlation kernel of the orthogonal polynomial ensemble tends to the sine kernel as the length |I|=2R tends to infinity, thereby answering a question posed by A.I. Bufetov.

Level repulsion for Schrödinger operators with singular continuous spectrum

We describe a family of half-line continuum Schrödinger operators with purely singular continuous spectrum on (0, \infty) , exhibiting asymptotic strong level repulsion (known as clock behavior). This follows from the convergence of the renormalized continuum Christoffel–Darboux kernel to the sine kernel.

Universality at Weak and Strong Non-Hermiticity Beyond the Elliptic Ginibre Ensemble

We consider non-Gaussian extensions of the elliptic Ginibre ensemble of complex non-Hermitian random matrices by fixing the trace $\operatorname{Tr}(XX^*)$ of the matrix $X$ with a hard or soft constraint. These ensembles have correlated matrix entries and non-determinantal joint densities of the complex eigenvalues. We study global and local bulk statistics in these ensembles, in particular in the limit of weak non-Hermiticity introduced by Fyodorov, Khoruzhenko and Sommers. Here, the support of the limiting measure collapses to the real line. This limit was motivated by physics applications and interpolates between the celebrated sine and Ginibre kernel. Our results constitute a first proof of universality of the interpolating kernel. Furthermore, in the limit of strong non-Hermiticity, where the support of the limiting measure remains an ellipse, we obtain local Ginibre statistics in the bulk of the spectrum.

On Expanding Neighborhoods of Local Universality of Gaussian Unitary Ensembles

The classical universality theorem states that the Christoffel–Darboux kernel of the Hermite polynomials scaled by a factor of \(1/\sqrt n\) tends to the sine kernel in local variables \(\tilde x,\tilde y\) in a neighborhood of a point \(x^*\in(-\sqrt 2,\sqrt 2)\)). This classical result is well known for \(\tilde x,\tilde y\in{K}\Subset\mathbb{R}\). In this paper, we show that this classical result remains valid for expanding compact sets K = K(n). An interesting phenomenon of admissible dependence of the expansion rate of compact sets K(n) on x* is established. For \(x^*\in(-\sqrt 2,\sqrt 2)\backslash\left\{0\right\}\)) and for x* = 0, there are different growth regimes of compact sets K(n). A transient regime is found.

Accurate Measurement in Doppler Radar Vital Sign Detection Based on Parameterized Demodulation

Utilizing Doppler radar to conduct noncontact vital sign detection has attracted growing interest in recent years. Aiming to extract the vital sign information from the baseband signal effectively and accurately, a novel signal processing method based on parameterized demodulation (PD) is proposed. To effectively characterize the baseband signal whose phase consists of two oscillating components (i.e., the respiration and heartbeat components), the proposed algorithm defines a demodulation operator with sine kernel functions and formulates the phase demodulation as a parameter optimization problem. To increase the computational efficiency of the algorithm, the parameters corresponding to the respiration and heartbeat components are estimated sequentially. Specifically, the respiration component is first estimated and removed from the phase of the baseband signal, and then, the heartbeat component is extracted from the residual signal. Compared with the existing methods, the proposed algorithm is free of the resolution problem in fast Fourier transform-based methods with a limited data length and can obtain accurate rate tracking in a noisy environment. Both simulated and experimental results are provided to demonstrate the advantages and effectiveness of the proposed method for accurate noncontact vital sign detection.

Bulk and soft-edge universality for singular values of products of Ginibre random matrices

Il a été démontré par Akemann, Ipsen et Kieburg que les valeurs singulières carrées de produits de $M$ matrices aléatoires rectangulaires composées de variables aléatoires complexes gaussiennes, suivent un processus ponctuel déterminantal, avec un noyau de corrélation qui admet une représentation en fonction des fonctions G de Meijer. Nous démontrons l’universalité des statistiques locales des valeurs singulières carrées, c’est-à-dire, l’universalité dans le bulk donné par le noyau sinus et l’universalité au bord donné par le noyau Airy. La preuve est basée sur l’analyse asymptotique de la représentation du noyau de corrélation comme une double intégrale de contour. Notre stratégie peut être généralisée à d’autres modèles de produits de matrices aléatoires introduits récemment, afin d’obtenir des résultats d’universalité similaires. Deux autres exemples sont considérés, le premier étant le produit de $M$ matrices de Ginibre et l’inverse de $K$ matrices de Ginibre, étudié par Forrester, le deuxième étant le produit de $M-1$ matrices de Ginibre avec une troncation d’une matrice unitaire, considéré par Kuijlaars et Stivigny.

The circular unitary ensemble and the Riemann zeta function: the microscopic landscape and a new approach to ratios

We show in this paper that after proper scalings, the characteristic polynomial of a random unitary matrix converges to a random analytic function whose zeros, which are on the real line, form a determinantal point process with sine kernel. Our scaling is performed at the so-called “microscopic” level, that is we consider the characteristic polynomial at points whose distance to 1 has order 1 / n. We prove that the rescaled characteristic polynomial does not even have a moment of order one, hence making the classical techniques of random matrix theory difficult to apply. In order to deal with this issue, we couple all the dimensions n on a single probability space, in such a way that almost sure convergence occurs when n goes to infinity. The strong convergence results in this setup provide us with a new approach to ratios: we are able to solve open problems about the limiting distribution of ratios of characteristic polynomials evaluated at points of the form $$\exp (2 i \pi \alpha /n)$$ and related objects (such as the logarithmic derivative). We also explicitly describe the dependence relation for the logarithm of the characteristic polynomial evaluated at several points on the microscopic scale. On the number theory side, inspired by the work by Keating and Snaith, we conjecture some new limit theorems for the value distribution of the Riemann zeta function on the critical line at the level of stochastic processes.

Large N-limit for random matrices with external source with three distinct eigenvalues

In this paper, we analyze the large N-limit for random matrix with external source with three distinct eigenvalues. And we confine ourselves in the Hermite case and the three distinct eigenvalues are [Formula: see text]. For the case [Formula: see text], we establish the universal behavior of local eigenvalue correlations in the limit [Formula: see text], which is known from unitarily invariant random matrix models. Thus, local eigenvalue correlations are expressed in terms of the sine kernel in the bulk and in terms of the Airy kernel at the edge of the spectrum. The result can be obtained by analyzing [Formula: see text] Riemann–Hilbert problem via nonlinear steepest decent method.

Strong Markov property of determinantal processes with extended kernels

Noncolliding Brownian motion (Dyson’s Brownian motion model with parameter β=2) and noncolliding Bessel processes are determinantal processes; that is, their space–time correlation functions are represented by determinants. Under a proper scaling limit, such as the bulk, soft-edge and hard-edge scaling limits, these processes converge to determinantal processes describing systems with an infinite number of particles. The main purpose of this paper is to show the strong Markov property of these limit processes, which are determinantal processes with the extended sine kernel, extended Airy kernel and extended Bessel kernel, respectively. We also determine the quasi-regular Dirichlet forms and infinite-dimensional stochastic differential equations associated with the determinantal processes.