Linear recurrences and asymptotic behavior of exponential sums of symmetric boolean functions

In this paper we give an improvement of the degree of the homogeneous linear recurrence with integer coefficients that exponential sums of symmetric Boolean functions satisfy. This improvement is tight. We also compute the asymptotic behavior of symmetric Boolean functions and provide a formula that allows us to determine if a symmetric boolean function is asymptotically not balanced. In particular, when the degree of the symmetric function is a power of two, then the exponential sum is much smaller than $2^n$.Comment: 18 pages, 3 figure

Rosenblatt distribution subordinated to gaussian random fields with long-range dependence

The Karhunen-Lo\`eve expansion and the Fredholm determinant formula are used to derive an asymptotic Rosenblatt-type distribution of a sequence of integrals of quadratic functions of Gaussian stationary random fields on R^d displaying long-range dependence. This distribution reduces to the usual Rosenblatt distribution when d=1. Several properties of this new distribution are obtained. Specifically, its series representation in terms of independent chi-squared random variables is given, the asymptotic behavior of the eigenvalues, its L\`evy-Khintchine representation, as well as its membership to the Thorin subclass of self-decomposable distributions. The existence and boundedness of its probability density is then a direct consequence.Comment: This paper has 40 pages and it has already been submitte

Gauge field theory approach to spin transport in a 2D electron gas

We discuss the Pauli Hamiltonian including the spin-orbit interaction within an U(1) x SU(2) gauge theory interpretation, where the gauge symmetry appears to be broken. This interpretation offers new insight into the problem of spin currents in the condensed matter environment, and can be extended to Rashba and Dresselhaus spin-orbit interactions. We present a few outcomes of the present formulation: i) it automatically leads to zero spin conductivity, in contrast to predictions of Gauge symmetric treatments, ii) a topological quantization condition leading to voltage quantization follows, and iii) spin interferometers can be conceived in which, starting from a arbitrary incoming unpolarized spinor, it is always possible to construct a perfect spin filtering condition.Comment: Invited contribution to Statphys conference, June 2009, Lviv (Ukraine

Equilibrium currents in a Corbino graphene ring

We address the description of a graphene Corbino disk in the context of a tight binding approach that includes both kinetic and Rashba spin-orbit coupling due to an external out-of-plane electric field. Persistent equilibrium currents are induced by an external magnetic field breaking time reversal symmetry. By direct diagonalization, we compute the spectrum and focus on the dispersion near the $K$ points at the Fermi level. The dispersion keenly reproduces that of a continuum model in spite of the complexity of the boundary conditions. We validate the assumptions of the continuum model in terms of predominant zig-zag boundaries conditions and weak sub-band coupling. The wave functions displaying the lowest transverse modes are obtained, showing the predominance of edge states with charge density at the zig-zag edges. The persistent charge currents, nevertheless, do not follow the traditional argument of current cancellation from levels below the Fermi level, and thus they depart in the tight-binding from those found in the continuum model.Comment: 8 pages, 6 figure

On a class of minimum contrast estimators for Gegenbauer random fields

The article introduces spatial long-range dependent models based on the fractional difference operators associated with the Gegenbauer polynomials. The results on consistency and asymptotic normality of a class of minimum contrast estimators of long-range dependence parameters of the models are obtained. A methodology to verify assumptions for consistency and asymptotic normality of minimum contrast estimators is developed. Numerical results are presented to confirm the theoretical findings.Comment: 23 pages, 8 figure