Collapse of 4D random geometries

We extend the analysis of the Backgammon model to an ensemble with a fixed number of balls and a fluctuating number of boxes. In this ensemble the model exhibits a first order phase transition analogous to the one in higher dimensional simplicial gravity. The transition relies on a kinematic condensation and reflects a crisis of the integration measure which is probably a part of the more general problem with the measure for functional integration over higher (d>2) dimensional Riemannian structures.Comment: 7 pages, Latex2e, 2 figures (.eps

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

Phase diagram of the mean field model of simplicial gravity

We discuss the phase diagram of the balls in boxes model, with a varying number of boxes. The model can be regarded as a mean-field model of simplicial gravity. We analyse in detail the case of weights of the form $p(q) = q^{-\beta}$, which correspond to the measure term introduced in the simplicial quantum gravity simulations. The system has two phases~: {\em elongated} ({\em fluid}) and {\em crumpled}. For $\beta\in (2,\infty)$ the transition between these two phases is first order, while for $\beta \in (1,2]$ it is continuous. The transition becomes softer when $\beta$ approaches unity and eventually disappears at $\beta=1$. We then generalise the discussion to an arbitrary set of weights. Finally, we show that if one introduces an additional kinematic bound on the average density of balls per box then a new {\em condensed} phase appears in the phase diagram. It bears some similarity to the {\em crinkled} phase of simplicial gravity discussed recently in models of gravity interacting with matter fields.Comment: 15 pages, 5 figure

Causal and homogeneous networks

Growing networks have a causal structure. We show that the causality strongly influences the scaling and geometrical properties of the network. In particular the average distance between nodes is smaller for causal networks than for corresponding homogeneous networks. We explain the origin of this effect and illustrate it using as an example a solvable model of random trees. We also discuss the issue of stability of the scale-free node degree distribution. We show that a surplus of links may lead to the emergence of a singular node with the degree proportional to the total number of links. This effect is closely related to the backgammon condensation known from the balls-in-boxes model.Comment: short review submitted to AIP proceedings, CNET2004 conference; changes in the discussion of the distance distribution for growing trees, Fig. 6-right change

Asymmetric correlation matrices: an analysis of financial data

We analyze the spectral properties of correlation matrices between distinct statistical systems. Such matrices are intrinsically non symmetric, and lend themselves to extend the spectral analyses usually performed on standard Pearson correlation matrices to the realm of complex eigenvalues. We employ some recent random matrix theory results on the average eigenvalue density of this type of matrices to distinguish between noise and non trivial correlation structures, and we focus on financial data as a case study. Namely, we employ daily prices of stocks belonging to the American and British stock exchanges, and look for the emergence of correlations between two such markets in the eigenvalue spectrum of their non symmetric correlation matrix. We find several non trivial results, also when considering time-lagged correlations over short lags, and we corroborate our findings by additionally studying the asymmetric correlation matrix of the principal components of our datasets.Comment: Revised version; 11 pages, 13 figure

Random matrix model for QCD_3 staggered fermions

We show that the lowest part of the eigenvalue density of the staggered fermion operator in lattice QCD_3 at small lattice coupling constant beta has exactly the same shape as in QCD_4. This observation is quite surprising, since universal properties of the QCD_3 Dirac operator are expected to be described by a non-chiral matrix model. We show that this effect is related to the specific nature of the staggered fermion discretization and that the eigenvalue density evolves towards the non-chiral random matrix prediction when beta is increased and the continuum limit is approached. We propose a two-matrix model with one free parameter which interpolates between the two limits and very well mimics the pattern of evolution with beta of the eigenvalue density of the staggered fermion operator in QCD_3.Comment: 8 pages 4 figure

Using a Fermionic Ensemble of Systems to Determine Excited States

We discuss a new numerical method for the determination of excited states of a quantum system using a generalization of the Feynman-Kac formula. The method relies on introducing an ensemble of non-interacting identical systems with a fermionic statistics imposed on the systems as a whole, and on determining the ground state of this fermionic ensemble by taking the large time limit of the Euclidean kernel. Due to the exclusion principle, the ground state of an $n$-system ensemble is realized by the set of individual systems occupying successively the $n$ lowest states, all of which can therefore be sampled in this way. To demonstrate how the method works, we consider a one-dimensional oscillator and a chain of harmonically coupled particles.Comment: 14 pages, Latex + 4 eps figure

Resonance parameters of the first 1/2+ state in 9Be and astrophysical implications

Spectra of the 9Be(e,e') reaction have been measured at the S-DALINAC at an electron energy E_0 = 73 MeV and scattering angles of 93{\deg} and 141{\deg} with high energy resolution up to excitation energies E_x = 8 MeV. The astrophysically relevant resonance parameters of the first excited 1/2+ state of 9Be have been extracted in a one-level approximation of R-matrix theory resulting in a resonance energy E_R = 1.748(6) MeV and width Gamma_R = 274(8) keV in good agreement with the latest 9Be(gamma,n) experiment but with considerably improved uncertainties. However, the reduced B(E1) transition strength deduced from an extrapolation of the (e,e') data to the photon point is a factor of two smaller. Implications of the new results for a possible production of 12C in neutron-rich astrophysical scenarios are discussed.Comment: 8 pages, 7 figures, accepted for publication in Phys. Rev.

Phase transitions in social networks

We study a model of network with clustering and desired node degree. The original purpose of the model was to describe optimal structures of scientific collaboration in the European Union. The model belongs to the family of exponential random graphs. We show by numerical simulations and analytical considerations how a very simple Hamiltonian can lead to surprisingly complicated and eventful phase diagram.Comment: 8 pages, 8 figure