Moment instabilities in multidimensional systems with noise

We present a systematic study of moment evolution in multidimensional stochastic difference systems, focusing on characterizing systems whose low-order moments diverge in the neighborhood of a stable fixed point. We consider systems with a simple, dominant eigenvalue and stationary, white noise. When the noise is small, we obtain general expressions for the approximate asymptotic distribution and moment Lyapunov exponents. In the case of larger noise, the second moment is calculated using a different approach, which gives an exact result for some types of noise. We analyze the dependence of the moments on the system's dimension, relevant system properties, the form of the noise, and the magnitude of the noise. We determine a critical value for noise strength, as a function of the unperturbed system's convergence rate, above which the second moment diverges and large fluctuations are likely. Analytical results are validated by numerical simulations. We show that our results cannot be extended to the continuous time limit except in certain special cases.Comment: 21 pages, 15 figure

Critical behavior of loops and biconnected clusters on fractals of dimension d < 2

We solve the O(n) model, defined in terms of self- and mutually avoiding loops coexisting with voids, on a 3-simplex fractal lattice, using an exact real space renormalization group technique. As the density of voids is decreased, the model shows a critical point, and for even lower densities of voids, there is a dense phase showing power-law correlations, with critical exponents that depend on n, but are independent of density. At n=-2 on the dilute branch, a trivalent vertex defect acts as a marginal perturbation. We define a model of biconnected clusters which allows for a finite density of such vertices. As n is varied, we get a line of critical points of this generalized model, emanating from the point of marginality in the original loop model. We also study another perturbation of adding local bending rigidity to the loop model, and find that it does not affect the universality class.Comment: 14 pages,10 figure

Universality and Critical Phenomena in String Defect Statistics

The idea of biased symmetries to avoid or alleviate cosmological problems caused by the appearance of some topological defects is familiar in the context of domain walls, where the defect statistics lend themselves naturally to a percolation theory description, and for cosmic strings, where the proportion of infinite strings can be varied or disappear entirely depending on the bias in the symmetry. In this paper we measure the initial configurational statistics of a network of string defects after a symmetry-breaking phase transition with initial bias in the symmetry of the ground state. Using an improved algorithm, which is useful for a more general class of self-interacting walks on an infinite lattice, we extend the work in \cite{MHKS} to better statistics and a different ground state manifold, namely $\R P^2$, and explore various different discretisations. Within the statistical errors, the critical exponents of the Hagedorn transition are found to be quite possibly universal and identical to the critical exponents of three-dimensional bond or site percolation. This improves our understanding of the percolation theory description of defect statistics after a biased phase transition, as proposed in \cite{MHKS}. We also find strong evidence that the existence of infinite strings in the Vachaspati Vilenkin algorithm is generic to all (string-bearing) vacuum manifolds, all discretisations thereof, and all regular three-dimensional lattices.Comment: 62 pages, plain LaTeX, macro mathsymb.sty included, figures included. also available on http://starsky.pcss.maps.susx.ac.uk/groups/pt/preprints/96/96011.ps.g

Two Ising Models Coupled to 2-Dimensional Gravity

To investigate the properties of $c=1$ matter coupled to $2$d{--}gravity we have performed large-scale simulations of two copies of the Ising Model on a dynamical lattice. We measure spin susceptibility and percolation critical exponents using finite-size scaling. We show explicitly how logarithmic corrections are needed for a proper comparison with theoretical exponents. We also exhibit correlations, mediated by gravity, between the energy and magnetic properties of the two Ising species. The prospects for extending this work beyond $c=1$ are addressed.Comment: revised version w/ typos corrected; standard latex w/ epsf and 9 figure

Finite Size Analysis of the U(1) Phase Transition using the World-sheet Formulation

We present a high statistics analysis of the pure gauge compact U(1) lattice theory using the the world-sheet or Lagrangian loop representation. We have applied a simulation method that deals directly with (gauge invariant) integer variables on plaquettes. As a result we get a significant amelioration of the simulation that allows to work with large lattices avoiding the metaestability problems that appear using the standard Wilson formulation.Comment: 14 pages, 4 figures. REVTEX and eps