Two-dimensional White Noise Research Articles

Integrated density of states of the Anderson Hamiltonian with two-dimensional white noise

We construct the integrated density of states of the Anderson Hamiltonian with two-dimensional white noise by proving the convergence of the Dirichlet eigenvalue counting measures associated with the Anderson Hamiltonians on the boxes. We also determine the logarithmic asymptotics of the left tail of the integrated density of states. Furthermore, we apply our result to a moment explosion of the parabolic Anderson model in the plane.

A stochastic telegraph equation from the six-vertex model

A stochastic telegraph equation is defined by adding a random inhomogeneity to the classical (second-order linear hyperbolic) telegraph differential equation. The inhomogeneities we consider are proportional to the two-dimensional white noise, and solutions to our equation are two-dimensional random Gaussian fields. We show that such fields arise naturally as asymptotic fluctuations of the height function in a certain limit regime of the stochastic six-vertex model in a quadrant. The corresponding law of large numbers—the limit shape of the height function—is described by the (deterministic) homogeneous telegraph equation.

2D-frequency domain identification of complex sinusoids in the presence of additive noise

This paper describes a new approach for identifying the parameters of two-dimensional complex sinusoids from a finite number of measurements, in presence of additive and uncorrelated two-dimensional white noise. The proposed approach is based on using frequency domain data. The new method extends to the two-dimensional (2D) case some recent results obtained with reference to the frequency ESPRIT algorithm. The properties of the proposed method are analyzed by means of Monte Carlo simulations and its features are compared with those of a classical time domain estimation algorithm. The practical advantages of the method are highlighted. In fact the novel approach can operate just on a specified sub-area of the 2D spectrum. This area-selective feature allows a drastic reduction of the computational complexity, which is usually very high when standard time domain methods are used.

Identification of two dimensional complex sinusoids in white noise: a state-space frequency approach

This paper proposes a new frequency domain approach for identifying the parameters of two-dimensional complex sinusoids from a finite number of data, when the measurements are affected by additive and uncorrelated two-dimensional white noise. The new method extends in two dimensions a frequency identification procedure of complex sinusoids, originally developed for the one-dimensional case. The properties of the proposed method are analyzed by means of Monte Carlo simulations and its features are compared with those of other estimation algorithms. In particular the practical advantage of the method is highlighted. In fact the novel approach can operate just on a specified sub-area of the 2D spectrum. This area-selective feature allows a drastic reduction of the computational complexity, which is usually very high when standard time domain methods are used.

Visibility graphs of random scalar fields and spatial data.

We extend the family of visibility algorithms to map scalar fields of arbitrary dimension into graphs, enabling the analysis of spatially extended data structures as networks. We introduce several possible extensions and provide analytical results on the topological properties of the graphs associated to different types of real-valued matrices, which can be understood as the high and low disorder limits of real-valued scalar fields. In particular, we find a closed expression for the degree distribution of these graphs associated to uncorrelated random fields of generic dimension. This result holds independently of the field's marginal distribution and it directly yields a statistical randomness test, applicable in any dimension. We showcase its usefulness by discriminating spatial snapshots of two-dimensional white noise from snapshots of a two-dimensional lattice of diffusively coupled chaotic maps, a system that generates high dimensional spatiotemporal chaos. The range of potential applications of this combinatorial framework includes image processing in engineering, the description of surface growth in material science, soft matter or medicine, and the characterization of potential energy surfaces in chemistry, disordered systems, and high energy physics. An illustration on the applicability of this method for the classification of the different stages involved in carcinogenesis is briefly discussed.

Randomized Hamiltonian Feynman integrals and Schrödinger-Itô stochastic equations

In this paper, we consider stochastic Schrödinger equations with two-dimensional white noise. Such equations are used to describe the evolution of an open quantum system undergoing a process of continuous measurement. Representations are obtained for solutions of such equations using a generalization to the stochastic case of the classical construction of Feynman path integrals over trajectories in the phase space.

Blowup for the heat equation with a noise term

In this paper we study blow up of the equation\(u_t = u_{xx} + u^\gamma \dot W_{tx}\), where\(\dot W_{tx}\) is a two-dimensional white noise field and where Dirichlet boundary conditions are enforced. It is known that if γ<3/2, then the solution exists for all time; in this paper we show that if γ is much larger than 3/2, then the solution blows up in finite time with positive probability. We prove this by considering how peaks in the solution propagate. If a peak of high mass forms, we rescale the equation and divide the mass of the peak into a collection of peaks of smaller mass, and these peaks evolve almost independently. In this way we compare the evolution ofu to a branching process. Large peaks are regarded as particles in this branching process. Offspring are peaks which are higher by some factor. We show that the expected number of offspring is greater than one when γ is much larger than 3/2, and thus the branching process survives with positive probability, corresponding to blowup in finite time.

Two-dimensional smoothing of a vector Laplacian random field with application to geodesy (Ph.D. Thesis abstr.)

A two-dimensional signal processing algorithm is developed to obtain smoothed estimates of a vector random field when observed in the presence of additive two-dimensional white noise. The vector random field consists of a two-dimensional scalar random field and its two orthogonal derivatives, each of which can be scaled arbitrarily by a constant. The signal model consists of the Laplacian operator acting on the two-dimensional scalar random field excited by two-dimensional white noise. Such a signal model that permits a nonstationary and nonisotropic random field is "spatially dynamic" as opposed to a static model specified by autocorrelation functions or power spectral densities. The model is also "bilateral" such that any given spatial point’s dependency upon all its neighboring points is preserved and no unnecessary limitation of causality is enforced on the two-dimensional system. To preserve the noncausality of the problem, the estimation algorithm must not order the two-dimensional data since there is no sense in which any data point can be said to be "prior" to any other data point. To this end, a two-dimensional smoothing algorithm is developed first in the continuous domain by making use of a two-dimensional Karhunen-Loeve expansion that makes no specific reference to either the geometry or the ordering of the parameter space.

The two-dimensional white noise problem and localisation in an inversion layer

The density of states in the two-dimensional white noise problem is calculated from the coherent potential approximation at high energy and from fluctuation theory at low energies; both the exponent and the power of E in the prefactor for the density of fluctuation states are evaluated. Comparison is made with the results of a tight binding simulation of the white noise problem, and good agreement between theory and calculation is obtained. Calculation of the conductivity of different sized samples enables the mobility edge to be determined, and it is found that the critical value of g=n(Ec)/n0 is more than 0.8, in contrast with much lower values suggested earlier. The metallic conductivity is compared with the CPA result, and found to be lower by a factor of 0.4. Some modifications are introduced into the theory when account is taken of orbital degeneracy. Comparison is made with Pollitt's experiments on the inversion layer.