Nicholas R. Moloney

Complexity and criticality

Percolation: Percolating Phase Transition In One and Two Dimensions, and in the Bethe Lattice Geometric Properties of Clusters Scaling Ansatz, Scaling Functions and Scaling Relations Universality Real-Space Renormalisation Group Ising Model: Review of Thermodynamics and Statistical Mechanics Symmetry Breaking Ferromagnetic Phase Transition In One and Two Dimensions, and in the Mean-Field Landau Theory of Continuous Phase Transitions Scaling Ansatz, Scaling Functions and Scaling Relations Universality Real-Space Renormalisation Group Self-Organised Criticality: BTW Model in One and Two Dimensions, and in the Mean-Field A Rice Pile Experiment and the Oslo Model Earthquakes and the OFC Model Rainfall Self-Organised Criticality as a Unifying Principle.

Data-driven prediction of thresholded time series of rainfall and self-organized criticality models

We study the occurrence of events, subject to threshold, in a representative self-organized criticality (SOC) sandpile model and in high-resolution rainfall data. The predictability in both systems is analyzed by means of a decision variable sensitive to event clustering, and the quality of the predictions is evaluated by the receiver operating characteristic (ROC) method. In the case of the SOC sandpile model, the scaling of quiet-time distributions with increasing threshold leads to increased predictability of extreme events. A scaling theory allows us to understand all the details of the prediction procedure and to extrapolate the shape of the ROC curves for the most extreme events. For rainfall data, the quiet-time distributions do not scale for high thresholds, which means that the corresponding ROC curves cannot be straightforwardly related to those for lower thresholds. In this way, ROC curves are useful for highlighting differences in predictability of extreme events between toy models and real-world phenomena.

The perils of thresholding

The thresholding of time series of activity or intensity is frequently used to define and differentiate events. This is either implicit, for example due to resolution limits, or explicit, in order to filter certain small scale physics from the supposed true asymptotic events. Thresholding the birth–death process, however, introduces a scaling region into the event size distribution, which is characterized by an exponent that is unrelated to the actual asymptote and is rather an artefact of thresholding. As a result, numerical fits of simulation data produce a range of exponents, with the true asymptote visible only in the tail of the distribution. This tail is increasingly difficult to sample as the threshold is increased. In the present case, the exponents and the spurious nature of the scaling region can be determined analytically, thus demonstrating the way in which thresholding conceals the true asymptote. The analysis also suggests a procedure for detecting the influence of the threshold by means of a data collapse involving the threshold-imposed scale.

Asynchronously parallelized percolation on distributed machines.

We propose a powerful method based on the Hoshen-Kopelman algorithm for simulating percolation asynchronously on distributed machines. Our method demands very little of hardware and yet we are able to make high precision measurements on very large lattices. We implement our method to calculate various cluster size distributions on large lattices of different aspect ratios spanning three orders of magnitude for two-dimensional site and bond percolation. We find that the nonuniversal constants in the scaling function for the cluster size distribution apparently satisfy a scaling relation, and that the moment ratios for the largest cluster size distribution reveal a characteristic aspect ratio at r approximately 9.

Numerical results for crossing, spanning and wrapping in two-dimensional percolation

Using a recently developed method to simulate percolation on large clusters of distributed machines [1], we have numerically calculated crossing, spanning and wrapping probabilities in two-dimensional site and bond percolation with exceptional accuracy. Our results are fully consistent with predictions from conformal field theory. We present many new results that await theoretical explanation, particularly for wrapping clusters on a cylinder. We therefore provide possibly the most up-to-date reference for theoreticians working on crossing, spanning and wrapping probabilities in two-dimensional percolation.

Winding Clusters in Percolation on the Torus and the Möbius Strip

Using a simulation technique introduced recently, we study winding clusters in percolation on the torus and the Möbius strip for different aspect ratios. The asynchronous parallelization of the simulation makes very large system and sample sizes possible. Our high accuracy results are fully consistent with predictions from conformal field theory. The numerical results for the Möbius strip and the number distribution of winding clusters on the torus await theoretical explanation. To our knowledge, this study is the first of its kind.

Avalanche behavior in an absorbing state Oslo model

Self-organized criticality can be translated into the language of absorbing state phase transitions. Most models for which this analogy is established have been investigated for their absorbing state characteristics. In this paper, we transform the self-organized critical Oslo model into an absorbing state Oslo model and analyze the avalanche behavior. We find that the resulting gap exponent D is consistent with its value in the self-organized critical model. For the avalanche size exponent tau an analysis of the effect of the external drive and the boundary conditions is required.

Percolation on trees as a Brownian excursion: From Gaussian to Kolmogorov-Smirnov to exponential statistics

We calculate the distribution of the size of the percolating cluster on a tree in the subcritical, critical, and supercritical phase. We do this by exploiting a mapping between continuum trees and Brownian excursions, and arrive at a diffusion equation with suitable boundary conditions. The exact solution to this equation can be conveniently represented as a characteristic function, from which the following distributions are clearly visible: Gaussian (subcritical), Kolmogorov-Smirnov (critical), and exponential (supercritical). In this way we provide an intuitive explanation for the result reported in Botet and Płoszajczak, Phys. Rev. Lett. 95, 185702 (2005)PRLTAO0031-900710.1103/PhysRevLett.95.185702 for critical percolation.

Criticality on Rainfall: Statistical Observational Constraints for the Onset of Strong Convection Modelling

A better understanding of convection is crucial for reducing the intrinsic errors present in climate models [4]. Many atmospheric processes related to precipitation have large scale correlations in time and space, which are the result of the coupling between several non-linear mechanisms with different temporal and spatial characteristic scales. Despite the diversity of individual rain events, a recent array of statistical measures presents surprising statistical regularities giving support to the hypothesis that atmospheric convection and precipitation may be a real-world example of Self-Organised Criticality (SOC) [2, 16].