Correlation functions and critical behaviour on fluctuating geometries

We study the two-point correlation function in the model of branched polymers and its relation to the critical behaviour of the model. We show that the correlation function has a universal scaling form in the generic phase with the only scale given by the size of the polymer. We show that the origin of the singularity of the free energy at the critical point is different from that in the standard statistical models. The transition is related to the change of the dimensionality of the system.Comment: 10 Pages, Latex2e, uses elsart.cls, 1 figure include

Signal and Noise in Correlation Matrix

Using random matrix technique we determine an exact relation between the eigenvalue spectrum of the covariance matrix and of its estimator. This relation can be used in practice to compute eigenvalue invariants of the covariance (correlation) matrix. Results can be applied in various problems where one experimentally estimates correlations in a system with many degrees of freedom, like in statistical physics, lattice measurements of field theory, genetics, quantitative finance and other applications of multivariate statistics.Comment: 17 pages, 3 figures, corrected typos, revtex style changed to elsar

Eigenvalue density of empirical covariance matrix for correlated samples

We describe a method to determine the eigenvalue density of empirical covariance matrix in the presence of correlations between samples. This is a straightforward generalization of the method developed earlier by the authors for uncorrelated samples. The method allows for exact determination of the experimental spectrum for a given covariance matrix and given correlations between samples in the limit of large N and N/T=r=const with N being the number of degrees of freedom and T being the number of samples. We discuss the effect of correlations on several examples.Comment: 12 pages, 5 figures, to appear in Acta Phys. Pol. B (Proceedings of the conference on `Applications of Random Matrix Theory to Economy and Other Complex Systems', May 25-28, 2005, Cracow, Polan

4D Quantum Gravity Coupled to Matter

We investigate the phase structure of four-dimensional quantum gravity coupled to Ising spins or Gaussian scalar fields by means of numerical simulations. The quantum gravity part is modelled by the summation over random simplicial manifolds, and the matter fields are located in the center of the 4-simplices, which constitute the building blocks of the manifolds. We find that the coupling between spin and geometry is weak away from the critical point of the Ising model. At the critical point there is clear coupling, which qualitatively agrees with that of gaussian fields coupled to gravity. In the case of pure gravity a transition between a phase with highly connected geometry and a phase with very ``dilute'' geometry has been observed earlier. The nature of this transition seems unaltered when matter fields are included. It was the hope that continuum physics could be extracted at the transition between the two types of geometries. The coupling to matter fields, at least in the form discussed in this paper, seems not to improve the scaling of the curvature at the transition point.Comment: 15 pages, 9 figures (available as PS-files by request). Late

Perturbing General Uncorrelated Networks

This paper is a direct continuation of an earlier work, where we studied Erd\"os-R\'enyi random graphs perturbed by an interaction Hamiltonian favouring the formation of short cycles. Here, we generalize these results. We keep the same interaction Hamiltonian but let it act on general graphs with uncorrelated nodes and an arbitrary given degree distribution. It is shown that the results obtained for Erd\"os-R\'enyi graphs are generic, at the qualitative level. However, scale-free graphs are an exception to this general rule and exhibit a singular behaviour, studied thoroughly in this paper, both analytically and numerically.Comment: 7 pages, 7 eps figures, 2-column revtex format, references adde