Re-examination of log-periodicity observed in the seismic precursors of the 1989 Loma Prieta earthquake

Based on several empirical evidence, a series of papers has advocated the concept that seismicity prior to a large earthquake can be understood in terms of the statistical physics of a critical phase transition. In this model, the cumulative Benioff strain (BS) increases as a power-law time-to-failure before the final event. This power law reflects a kind of scale invariance with respect to the distance to the critical point. A few years ago, on the basis of a fit of the cumulative BS released prior to the 1989 Loma Prieta earthquake, Sornette and Sammis [1995] proposed that this scale invariance could be partially broken into a discrete scale invariance (DSI). The observable consequence of DSI takes the form of log-periodic oscillations decorating the accelerating power law. They found that the quality of the fit and the predicted time of the event are significantly improved by the introduction of log-periodicity. Here, we present a battery of synthetic tests performed to quantify the statistical significance of this claim. We find that log-periodic oscillations with frequency and regularity similar to those of the Loma Prieta case are very likely to be generated by the interplay of the low pass filtering step due to the construction of cumulative functions together with the approximate power law acceleration. Thus, the single Loma Prieta case alone cannot support the initial claim and additional cases and further study are needed to increase the signal-to-noise ratio if any. The present study will be a useful methodological benchmark for future testing of additional events when the methodology and data to construct reliable Benioff strain function become available.Comment: LaTeX, JGR preprint with AGU++ v16.b and AGUTeX 5.0, use packages graphicx and psfrag, 23 eps figures, 17 pages. In press J. Geophys. Re

Artifactual log-periodicity in finite size data: Relevance for earthquake aftershocks

The recently proposed discrete scale invariance and its associated log-periodicity are an elaboration of the concept of scale invariance in which the system is scale invariant only under powers of specific values of the magnification factor. We report on the discovery of a novel mechanism for such log-periodicity relying solely on the manipulation of data. This ``synthetic'' scenario for log-periodicity relies on two steps: (1) the fact that approximately logarithmic sampling in time corresponds to uniform sampling in the logarithm of time; and (2) a low-pass-filtering step, as occurs in constructing cumulative functions, in maximum likelihood estimations, and in de-trending, reddens the noise and, in a finite sample, creates a maximum in the spectrum leading to a most probable frequency in the logarithm of time. We explore in detail this mechanism and present extensive numerical simulations. We use this insight to analyze the 27 best aftershock sequences studied by Kisslinger and Jones [1991] to search for traces of genuine log-periodic corrections to Omori's law, which states that the earthquake rate decays approximately as the inverse of the time since the last main shock. The observed log-periodicity is shown to almost entirely result from the ``synthetic scenario'' owing to the data analysis. From a statistical point of view, resolving the issue of the possible existence of log-periodicity in aftershocks will be very difficult as Omori's law describes a point process with a uniform sampling in the logarithm of the time. By construction, strong log-periodic fluctuations are thus created by this logarithmic sampling.Comment: LaTeX, JGR preprint with AGU++ v16.b and AGUTeX 5.0, use packages graphicx, psfrag and latexsym, 41 eps figures, 26 pages. In press J. Geophys. Re

Effect of interactions on the noise of chiral Luttinger liquid systems

We analyze the current noise, generated at a quantum point contact in fractional quantum Hall edge state devices, using the chiral Luttinger liquid model with an impurity and the associated exact field theoretic solution. We demonstrate that an experimentally relevant regime of parameters exists where the noise coincides with the partition noise of independent Laughlin quasiparticles. However, outside of this regime, this independent particle picture breaks down and the inclusion of interaction effects is essential to understand the shot noise.Comment: 4 pages, 3 figures; v2: modified FIG.1, new FIG.

A nested sequence of projectors (2): Multiparameter multistate statistical models, Hamiltonians, S-matrices

Our starting point is a class of braid matrices, presented in a previous paper, constructed on a basis of a nested sequence of projectors. Statistical models associated to such $N^2\times N^2$ matrices for odd $N$ are studied here. Presence of $\frac 12(N+3)(N-1)$ free parameters is the crucial feature of our models, setting them apart from other well-known ones. There are $N$ possible states at each site. The trace of the transfer matrix is shown to depend on $\frac 12(N-1)$ parameters. For order $r$, $N$ eigenvalues consitute the trace and the remaining $(N^r-N)$ eigenvalues involving the full range of parameters come in zero-sum multiplets formed by the $r$-th roots of unity, or lower dimensional multiplets corresponding to factors of the order $r$ when $r$ is not a prime number. The modulus of any eigenvalue is of the form $e^{\mu\theta}$, where $\mu$ is a linear combination of the free parameters, $\theta$ being the spectral parameter. For $r$ a prime number an amusing relation of the number of multiplets with a theorem of Fermat is pointed out. Chain Hamiltonians and potentials corresponding to factorizable $S$-matrices are constructed starting from our braid matrices. Perspectives are discussed.Comment: 32 pages, no figure, few mistakes are correcte

Stochastics theory of log-periodic patterns

We introduce an analytical model based on birth-death clustering processes to help understanding the empirical log-periodic corrections to power-law scaling and the finite-time singularity as reported in several domains including rupture, earthquakes, world population and financial systems. In our stochastics theory log-periodicities are a consequence of transient clusters induced by an entropy-like term that may reflect the amount of cooperative information carried by the state of a large system of different species. The clustering completion rates for the system are assumed to be given by a simple linear death process. The singularity at t_{o} is derived in terms of birth-death clustering coefficients.Comment: LaTeX, 1 ps figure - To appear J. Phys. A: Math & Ge

Numerical Evaluation of Shot Noise using Real Time Simulations

We present a method to determine the shot noise in quantum systems from knowledge of their time evolution - the latter being obtained using numerical simulation techniques. While our ultimate goal is the study of interacting systems, the main issues for the numerical determination of the noise do not depend on the interactions. To discuss them, we concentrate on the single resonant level model, which consists in a single impurity attached to non-interacting leads, with spinless fermions. We use exact diagonalisations (ED) to obtain time evolution, and are able to use known analytic results as benchmarks. We obtain a complete characterization of finite size effects at zero frequency, where we find that the finite size corrections scale $\propto G^2$, $G$ the differential conductance. We also discuss finite frequency noise, as well as the effects of damping in the leads.Comment: 6 pages, 7 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

D-brane interactions in type IIB plane-wave background

The cylinder diagrams that determine the static interactions between pairs of Dp-branes in the type IIB plane wave background are evaluated. The resulting expressions are elegant generalizations of the flat-space formulae that depend on the value of the Ramond-Ramond flux of the background in a non-trivial manner. The closed-string and open-string descriptions consistently transform into each other under a modular transformation only when each of the interacting D-branes separately preserves half the supersymmetries. These results are derived for configurations of euclidean signature D(p+1)-instantons but also generalize to lorentzian signature Dp-branes.Comment: 24 pages, Normalisation of boundary states correcte

Percolation in the Harmonic Crystal and Voter Model in three dimensions

We investigate the site percolation transition in two strongly correlated systems in three dimensions: the massless harmonic crystal and the voter model. In the first case we start with a Gibbs measure for the potential, $U=\frac{J}{2} \sum_{} (\phi(x) - \phi(y))^2$, $x,y \in \mathbb{Z}^3$, $J > 0$ and $\phi(x) \in \mathbb{R}$, a scalar height variable, and define occupation variables $\rho_h(x) =1,(0)$ for $\phi(x) > h (<h)$. The probability $p$ of a site being occupied, is then a function of $h$. In the voter model we consider the stationary measure, in which each site is either occupied or empty, with probability $p$. In both cases the truncated pair correlation of the occupation variables, $G(x-y)$, decays asymptotically like $|x-y|^{-1}$. Using some novel Monte Carlo simulation methods and finite size scaling we find accurate values of $p_c$ as well as the critical exponents for these systems. The latter are different from that of independent percolation in $d=3$, as expected from the work of Weinrib and Halperin [WH] for the percolation transition of systems with $G(r) \sim r^{-a}$ [A. Weinrib and B. Halperin, Phys. Rev. B 27, 413 (1983)]. In particular the correlation length exponent $\nu$ is very close to the predicted value of 2 supporting the conjecture by WH that $\nu= \frac{2}{a}$ is exact.Comment: 8 figures. new version significantly different from the old one, includes new results, figures et

Interplay of the Scaling Limit and the Renormalization Group: Implications for Symmetry Restoration

Symmetry restoration is usually understood as a renormalization group induced phenomenon. In this context, the issue of whether one-loop RG equations can be trusted in predicting symmetry restoration has recently been the subject of much debate. Here we advocate a more pragmatic point of view and expand the definition of symmetry restoration to encompass all situations where the physical properties have only a weak dependence upon an anisotropy in the bare couplings. Moreover we concentrate on universal properties, and so take a scaling limit where the physics is well described by a field theory. In this context, we find a large variety of models that exhibit, for all practical purposes, symmetry restoration: even if symmetry is not restored in a strict sense, physical properties are surprisingly insensitive to the remaining anisotropy. Although we have adopted an expanded notion of symmetry restoration, we nonetheless emphasize that the scaling limit also has implications for symmetry restoration as a renormalization group induced phenomenon. In all the models we considered, the scaling limit turns out to only permit bare couplings which are nearly isotropic and small. Then the one-loop beta-function should contain all the physics and higher loop orders can be neglected. We suggest that this feature generalizes to more complex models. We exhibit a large class of theories with current-current perturbations (of which the SO(8) model of interest in two-leg Hubbard ladders/armchair carbon nanotubes is one) where the one-loop beta-functions indicates symmetry restoration and so argue that these results can be trusted within the scaling limit.Comment: 20 pages, 11 figures, RevTe