arXiv subjects

P. Grassberger

Publications and source records attributed to P. Grassberger.

At least 19 recordsLinked to original sources

arXiv2026

Strongly-connected percolation on directed lattices

We study percolation on lattices with directed bonds, focusing on the behavior of strongly-connected percolation clusters -- clusters in which every site is reachable from every other along a directed path. We consider the two-dimensional square lattice and various globally isotropic arrangements of the directions of the bonds. Performing simulations using a range of algorithmic approaches, we calculate high-precision values for critical exponents, fractal dimensions, crossing probabilities, and percolation thresholds for bond percolation with each bond arrangement. We find that the critical behavior is in a distinctly different universality class from that of traditional undirected percolation, but that all bond arrangements appear to fall in the same universality class.

cond-mat.stat-mech↗

arXiv2017

Universality of critically pinned interfaces in 2-dimensional isotropic random media

Based on extensive simulations, we conjecture that critically pinned interfaces in 2-dimensional isotropic random media with short range correlations are always in the universality class of ordinary percolation. Thus, in contrast to interfaces in $>2$ dimensions, there is no distinction between fractal (i.e., percolative) and rough but non-fractal interfaces. Our claim includes interfaces in zero-temperature random field Ising models (both with and without spontaneous nucleation), in heterogeneous bootstrap percolation, and in susceptible-weakened-infected-removed (SWIR) epidemics. It does not include models with long range correlations in the randomness, and models where overhangs are explicitly forbidden (which would imply non-isotropy of the medium).

cond-mat.dis-nn↗

arXiv2014

Recent advances and open challenges in percolation

Percolation is the paradigm for random connectivity and has been one of the most applied statistical models. With simple geometrical rules a transition is obtained which is related to magnetic models. This transition is, in all dimensions, one of the most robust continuous transitions known. We present a very brief overview of more than 60 years of work in this area and discuss several open questions for a variety of models, including classical, explosive, invasion, bootstrap, and correlated percolation.

cond-mat.stat-mech↗

arXiv2012

Comment on "Dynamic Opinion Model and Invasion Percolation"

In J. Shao et al., PRL 103, 108701 (2009) the authors claim that a model with majority rule coarsening exhibits in d=2 a percolation transition in the universality class of invasion percolation with trapping. In the present comment we give compelling evidence, including high statistics simulations on much larger lattices, that this is not correct. and that the model is trivially in the ordinary percolation universality class.

physics.data-an↗

arXiv2012

Corrections to Scaling for Watersheds, Optimal Path Cracks, and Bridge Lines

We study the corrections to scaling for the mass of the watershed, the bridge line, and the optimal path crack in two and three dimensions. We disclose that these models have numerically equivalent fractal dimensions and leading correction-to-scaling exponents. We conjecture all three models to possess the same fractal dimension, namely, $d_f=1.2168\pm0.0005$ in 2D and $d_f=2.487\pm0.003$ in 3D, and the same exponent of the leading correction, $Ω=0.9\pm0.1$ and $Ω=1.0\pm0.1$, respectively. The close relations between watersheds, optimal path cracks in the strong disorder limit, and bridge lines are further supported by either heuristic or exact arguments.

cond-mat.stat-mech↗

arXiv2008

Networks of Recurrent Events, a Theory of Records, and an Application to Finding Causal Signatures in Seismicity

We propose a method to search for signs of causal structure in spatiotemporal data making minimal a priori assumptions about the underlying dynamics. To this end, we generalize the elementary concept of recurrence for a point process in time to recurrent events in space and time. An event is defined to be a recurrence of any previous event if it is closer to it in space than all the intervening events. As such, each sequence of recurrences for a given event is a record breaking process. This definition provides a strictly data driven technique to search for structure. Defining events to be nodes, and linking each event to its recurrences, generates a network of recurrent events. Significant deviations in properties of that network compared to networks arising from random processes allows one to infer attributes of the causal dynamics that generate observable correlations in the patterns. We derive analytically a number of properties for the network of recurrent events composed by a random process. We extend the theory of records to treat not only the variable where records happen, but also time as continuous. In this way, we construct a fully symmetric theory of records leading to a number of new results. Those analytic results are compared to the properties of a network synthesized from earthquakes in Southern California. Significant disparities from the ensemble of acausal networks that can be plausibly attributed to the causal structure of seismicity are: (1) Invariance of network statistics with the time span of the events considered, (2) Appearance of a fundamental length scale for recurrences, independent of the time span of the catalog, which is consistent with observations of the ``rupture length'', (3) Hierarchy in the distances and times of subsequent recurrences.

physics.data-an↗

arXiv2008

Entropy Estimates from Insufficient Samplings

We present a detailed derivation of some estimators of Shannon entropy for discrete distributions. They hold for finite samples of N points distributed into M "boxes", with N and M -> oo, but N/M < oo. In the high sampling regime (<< 1 points in each box) they have exponentially small biases. In the low sampling regime the errors increase but are still much smaller than for most other estimators. One advantage is that our main estimators are given analytically, with explicitly known analytical formulas for the biases.

physics.data-an↗

arXiv2002

Reply to ``Comments on Kullback-Leibler and renormalized entropies: Applications to electroencephalograms of epilepsy patients"

Kopitzki et al (preceeding comment) claim that the relationship between Renormalized and Kullback-Leibler entropies has already been given in their previous papers. Moreover, they argue that the first can give more useful information for e.g. localizing the seizure-generating area in epilepsy patients. In our reply we stress that if the relationship between both entropies would have been known by them, they should have noticed that the condition on the effective temperature is unnecessary. Indeed, this condition led them to choose different reference segments for different channels, even if this was physiologically unplausible. Therefore, we still argue that it is very unlikely that renormalized entropy will give more information than the conventional Kullback-Leibler entropy.

cond-mat.stat-mech↗

arXiv2002

Heat Conduction in Low Dimensions: From Fermi-Pasta-Ulam Chains to Single-Walled Nanotubes

Heat conduction in 1-dimensional anharmonic systems is anomalous in the sense that the conductivity κscales with a positive power of the system size, κ~ L^α. In two dimensions, previous simulations and theoretical arguments gave a logarithmic divergence. For rectangular systems of size L_\| x L_\perp there should be a cross-over from the 2-d to the 1-d behaviour as the aspect ratio r = L_\| / L_\perp increases from r=1 to r >> 1. When taking periodic boundary conditions in the transverse direction, this should be of direct relevance for the heat conduction in single-walled carbon nanotubes. In particular, one expects that k nanotubes of diameter R should conduct heat better than a single nanotube of the same length and of radius kR. We study this cross-over numerically by simulating the Fermi-Pasta-Ulam model. Apart from giving a precise estimate of the exponent α, our most intriguing results are that the divergence does not seem to be logarithmic in d=2 but also power-like, and that the cross-over does not happen at a fixed aspect ratio. Instead, it happens at r=r^* with r^* -> \infty for L -> \infty.

cond-mat.stat-mech↗

arXiv2002

Event synchronization: a simple and fast method to measure synchronicity and time delay patterns

We propose a simple method to measure synchronization and time delay patterns between signals. It is based on the relative timings of events in the time series, defined e.g. as local maxima. The degree of synchronization is obtained from the number of quasi-simultaneous appearances of events, and the delay is calculated from the precedence of events in one signal with respect to the other. Moreover, we can easily visualize the time evolution of the delay and synchronization level with an excellent resolution. We apply the algorithm to short rat EEG signals, some of them containing spikes. We also apply it to an intracranial human EEG recording containing an epileptic seizure, and we propose that the method might be useful for the detection of foci and for seizure prediction. It can be easily extended to other types of data and it is very simple and fast, thus being suitable for on-line implementations.

nlin.CD↗

arXiv2002

Go with the Winners: a General Monte Carlo Strategy

We describe a general strategy for sampling configurations from a given distribution, NOT based on the standard Metropolis (Markov chain) strategy. It uses the fact that nontrivial problems in statistical physics are high dimensional and often close to Markovian. Therefore, configurations are built up in many, usually biased, steps. Due to the bias, each configuration carries its weight which changes at every step. If the bias is close to optimal, all weights are similar and importance sampling is perfect. If not, ``population control" is applied by cloning/killing partial configurations with too high/low weight. This is done such that the final (weighted) distribution is unbiased. We apply this method (which is also closely related to diffusion type quantum Monte Carlo) to several problems of polymer statistics, reaction-diffusion models, sequence alignment, and percolation.

cond-mat.stat-mech↗

arXiv2001

On the performance of different synchronization measures in real data: a case study on EEG signals

We study the synchronization between left and right hemisphere rat EEG channels by using various synchronization measures, namely non-linear interdependences, phase-synchronizations, mutual information, cross-correlation and the coherence function. In passing we show a close relation between two recently proposed phase synchronization measures and we extend the definition of one of them. In three typical examples we observe that except mutual information, all these measures give a useful quantification that is hard to be guessed beforehand from the raw data. Despite their differences, results are qualitatively the same. Therefore, we claim that the applied measures are valuable for the study of synchronization in real data. Moreover, in the particular case of EEG signals their use as complementary variables could be of clinical relevance.

nlin.CD↗

arXiv2000

"Go with the winners"-Simulations

We describe a general strategy for sampling configurations from a given (Gibbs-Boltzmann or other) distribution. It is {\it not} based on the Metropolis concept of establishing a Markov process whose stationary state is the wanted distribution. Instead, it builds weighted instances according to a biased distribution. If the bias is optimal, all weights are equal and importance sampling is perfect. If not, "population control" is applied by cloning/killing configurations with too high/low weight. It uses the fact that nontrivial problems in statistical physics are high dimensional. Therefore, instances are built up in many steps, and the final weight can be guessed at an early stage. In contrast to evolutionary algorithms, the cloning/killing is done such that the wanted distribution is strictly observed without simultaneously keeping a large population in computer memory. We apply this method (which is also closely related to diffusion type quantum Monte Carlo) to several problems of polymer statistics, population dynamics, and percolation.

cond-mat.stat-mech↗

arXiv1999

Learning Driver-Response Relationships from Synchronization Patterns

We test recent claims that causal (driver/response) relationships can be deduced from interdependencies between simultaneously measured time series. We apply two recently proposed interdependence measures which should give similar results as cross predictabilities used by previous authors. The systems which we study are asymmetrically coupled simple models (Lorenz, Roessler, and Henon models), the couplings being such as to lead to generalized synchronization. If the data were perfect (noisefree, infinitely long), we should be able to detect, at least in some cases, which of the coupled systems is the driver and which the response. This might no longer be true if the time series has finite length. Instead, estimated interdependencies and mutual cross predictabilities depend strongly on which of the systems has a higher effective dimension at the typical neighborhood sizes used to estimate them, and causal relationships are more difficult to detect. We also show that slightly different variants of the interdependence measure can have quite different sensitivities.

chao-dyn↗

arXiv1999

Kullback-Leibler and Renormalized Entropy: Applications to EEGs of Epilepsy Patients

Recently, renormalized entropy was proposed as a novel measure of relative entropy (P. Saparin et al., Chaos, Solitons & Fractals 4, 1907 (1994)) and applied to several physiological time sequences, including EEGs of patients with epilepsy. We show here that this measure is just a modified Kullback-Leibler (K-L) relative entropy, and it gives similar numerical results to the standard K-L entropy. The latter better distinguishes frequency contents of e.g. seizure and background EEGs than renormalized entropy. We thus propose that renormalized entropy might not be as useful as claimed by its proponents. In passing we also make some critical remarks about the implementation of these methods.

physics.bio-ph↗

arXiv1999

Scaling of waves in the Bak-Tang-Wiesenfeld sandpile model

We study probability distributions of waves of topplings in the Bak-Tang-Wiesenfeld model on hypercubic lattices for dimensions D>=2. Waves represent relaxation processes which do not contain multiple toppling events. We investigate bulk and boundary waves by means of their correspondence to spanning trees, and by extensive numerical simulations. While the scaling behavior of avalanches is complex and usually not governed by simple scaling laws, we show that the probability distributions for waves display clear power law asymptotic behavior in perfect agreement with the analytical predictions. Critical exponents are obtained for the distributions of radius, area, and duration, of bulk and boundary waves. Relations between them and fractal dimensions of waves are derived. We confirm that the upper critical dimension D_u of the model is 4, and calculate logarithmic corrections to the scaling behavior of waves in D=4. In addition we present analytical estimates for bulk avalanches in dimensions D>=4 and simulation data for avalanches in D<=3. For D=2 they seem not easy to interpret.

cond-mat.stat-mech↗

arXiv1999

A Robust Method for Detecting Interdependences: Application to Intracranially Recorded EEG

We present a measure for characterizing statistical relationships between two time sequences. In contrast to commonly used measures like cross-correlations, coherence and mutual information, the proposed measure is non-symmetric and provides information about the direction of interdependence. It is closely related to recent attempts to detect generalized synchronization. However, we do not assume a strict functional relationship between the two time sequences and try to define the measure so as to be robust against noise, and to detect also weak interdependences. We apply our measure to intracranially recorded electroencephalograms of patients suffering from severe epilepsies.

chao-dyn↗

arXiv1999

Does macroscopic disorder imply microscopic chaos?

We argue that Gaspard and coworkers [Nature 394, 865 (1998)] do not give evidence for microscopic chaos in the sense in which they use the term. The effectively infinite number of molecules in a fluid can generate the same macroscopic disorder without any intrinsic instability. But we argue also that the notion of chaos in infinitely extended systems needs clarification: In a wider sense, even some systems without local instabilities can be considered chaotic.

cond-mat.stat-mech↗