Spectral properties of distance matrices

Distance matrices are matrices whose elements are the relative distances between points located on a certain manifold. In all cases considered here all their eigenvalues except one are non-positive. When the points are uncorrelated and randomly distributed we investigate the average density of their eigenvalues and the structure of their eigenfunctions. The spectrum exhibits delocalized and strongly localized states which possess different power-law average behaviour. The exponents depend only on the dimensionality of the manifold.Comment: 31 pages, 9 figure

Integration over connections in the discretized gravitational functional integrals

The result of performing integrations over connection type variables in the path integral for the discrete field theory may be poorly defined in the case of non-compact gauge group with the Haar measure exponentially growing in some directions. This point is studied in the case of the discrete form of the first order formulation of the Einstein gravity theory. Here the result of interest can be defined as generalized function (of the rest of variables of the type of tetrad or elementary areas) i. e. a functional on a set of probe functions. To define this functional, we calculate its values on the products of components of the area tensors, the so-called moments. The resulting distribution (in fact, probability distribution) has singular ($\delta$-function-like) part with support in the nonphysical region of the complex plane of area tensors and regular part (usual function) which decays exponentially at large areas. As we discuss, this also provides suppression of large edge lengths which is important for internal consistency, if one asks whether gravity on short distances can be discrete. Some another features of the obtained probability distribution including occurrence of the local maxima at a number of the approximately equidistant values of area are also considered.Comment: 22 page

The Berry Phase and Monopoles in Non-Abelian Gauge Theories

We consider the quantum mechanical notion of the geometrical (Berry) phase in SU(2) gauge theory, both in the continuum and on the lattice. It is shown that in the coherent state basis eigenvalues of the Wilson loop operator naturally decompose into the geometrical and dynamical phase factors. Moreover, for each Wilson loop there is a unique choice of U(1) gauge rotations which do not change the value of the Berry phase. Determining this U(1) locally in terms of infinitesimal Wilson loops we define monopole-like defects and study their properties in numerical simulations on the lattice. The construction is gauge dependent, as is common for all known definitions of monopoles. We argue that for physical applications the use of the Lorenz gauge is most appropriate. And, indeed, the constructed monopoles have the correct continuum limit in this gauge. Physical consequences are briefly discussed.Comment: 18 pp., Latex2e, 4 figures, psfig.st

Landau's necessary density conditions for LCA groups

H. Landau's necessary density conditions for sampling and interpolation may be viewed as a general principle resting on a basic fact of Fourier analysis: The complex exponentials $e^{i kx}$ ($k$ in $\mathbb{Z}$) constitute an orthogonal basis for $L^2([-\pi,\pi])$. The present paper extends Landau's conditions to the setting of locally compact abelian (LCA) groups, relying in an analogous way on the basics of Fourier analysis. The technicalities--in either case of an operator theoretic nature--are however quite different. We will base our proofs on the comparison principle of J. Ramanathan and T. Steger

Anisotropic KPZ growth in 2+1 dimensions: fluctuations and covariance structure

In [arXiv:0804.3035] we studied an interacting particle system which can be also interpreted as a stochastic growth model. This model belongs to the anisotropic KPZ class in 2+1 dimensions. In this paper we present the results that are relevant from the perspective of stochastic growth models, in particular: (a) the surface fluctuations are asymptotically Gaussian on a sqrt(ln(t)) scale and (b) the correlation structure of the surface is asymptotically given by the massless field.Comment: 13 pages, 4 figure