Distribution of sizes of erased loops of loop-erased random walks in two and three dimensions

We show that in the loop-erased random walk problem, the exponent characterizing probability distribution of areas of erased loops is superuniversal. In d-dimensions, the probability that the erased loop has an area A varies as A^{-2} for large A, independent of d, for 2 <= d <= 4. We estimate the exponents characterizing the distribution of perimeters and areas of erased loops in d = 2 and 3 by large-scale Monte Carlo simulations. Our estimate of the fractal dimension z in two-dimensions is consistent with the known exact value 5/4. In three-dimensions, we get z = 1.6183 +- 0.0004. The exponent for the distribution of durations of avalanche in the three-dimensional abelian sandpile model is determined from this by using scaling relations.Comment: 25 pages, 1 table, 8 figure

Dynamical properties of the Zhang model of Self-Organized Criticality

Critical exponents of the infinitely slowly driven Zhang model of self-organized criticality are computed for $d=2,3$ with particular emphasis devoted to the various roughening exponents. Besides confirming recent estimates of some exponents, new quantities are monitored and their critical exponents computed. Among other results, it is shown that the three dimensional exponents do not coincide with the Bak, Tang, and Wiesenfeld (abelian) model and that the dynamical exponent as computed from the correlation length and from the roughness of the energy profile do not necessarily coincide as it is usually implicitly assumed. An explanation for this is provided. The possibility of comparing these results with those obtained from Renormalization Group arguments is also briefly addressed.Comment: 8 pages, 12 PostScript figures, RevTe

Universality, frustration and conformal invariance in two-dimensional random Ising magnets

We consider long, finite-width strips of Ising spins with randomly distributed couplings. Frustration is introduced by allowing both ferro- and antiferromagnetic interactions. Free energy and spin-spin correlation functions are calculated by transfer-matrix methods. Numerical derivatives and finite-size scaling concepts allow estimates of the usual critical exponents $\gamma/\nu$, $\alpha/\nu$ and $\nu$ to be obtained, whenever a second-order transition is present. Low-temperature ordering persists for suitably small concentrations of frustrated bonds, with a transition governed by pure--Ising exponents. Contrary to the unfrustrated case, subdominant terms do not fit a simple, logarithmic-enhancement form. Our analysis also suggests a vertical critical line at and below the Nishimori point. Approaching this point along either the temperature axis or the Nishimori line, one finds non-diverging specific heats. A percolation-like ratio $\gamma/\nu$ is found upon analysis of the uniform susceptibility at the Nishimori point. Our data are also consistent with frustration inducing a breakdown of the relationship between correlation-length amplitude and critical exponents, predicted by conformal invariance for pure systems.Comment: RevTeX code for 10 pages, 9 eps figures, to appear in Physical Review B (September 1999

Self-Organized Criticality and Thermodynamic formalism

We introduce a dissipative version of the Zhang's model of Self-Organized Criticality, where a parameter allows to tune the local energy dissipation. We analyze the main dynamical features of the model and relate in particular the Lyapunov spectrum with the transport properties in the stationary regime. We develop a thermodynamic formalism where we define formal Gibbs measure, partition function and pressure characterizing the avalanche distributions. We discuss the infinite size limit in this setting. We show in particular that a Lee-Yang phenomenon occurs in this model, for the only conservative case. This suggests new connexions to classical critical phenomena.Comment: 35 pages, 15 Figures, submitte

Fluctuations of Spatial Patterns as a Measure of Classical Chaos

In problems where the temporal evolution of a nonlinear system cannot be followed, a method for studying the fluctuations of spatial patterns has been developed. That method is applied to well-known problems in deterministic chaos (the logistic map and the Lorenz model) to check its effectiveness in characterizing the dynamical behaviors. It is found that the indices $\mu _q$ are as useful as the Lyapunov exponents in providing a quantitative measure of chaos.Comment: 10 pages + 7 figures (in ps file), LaTex, Submitted to Phys. Rev.