The dynamical properties of the invasion percolation on the square lattice are investigated with an emphasis on the geometrical properties on the growing cluster of infected sites. The exterior frontier of this cluster forms a critical loop ensemble (CLE), whose length (l), the radius (r) and also roughness (w) fulfill the finite-size scaling hypothesis. The dynamical fractal dimension of the CLE defined as the exponent of the scaling relation between l and r is estimated to be Df = 1.76 ± 0.04. By studying the autocorrelation functions of these quantities we show importantly that there is a crossover between two time regimes, in which these functions change behavior from power-law at the small times, to exponential decay at long times. In the vicinity of this crossover time, these functions are estimated by log-normal functions. We also show that the increments of the considered statistical quantities, which are related to the random forces governing the dynamics of the observables undergo an anticorrelation/correlation transition at the time that the crossover takes place.

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Using the method developed in a recent paper (2019 Euro. Phys. J. B 92 1–28) we consider 1/f noise in two-dimensional electron gas (2DEG). The electron coherence length of the system is considered as a basic parameter for discretizing the space, inside which the dynamics of electrons is described by quantum mechanics, while for length scales much larger than it the dynamics is semi-classical. For our model, which is based on the Thomas-Fermi–Dirac approximation, there are two control parameters: temperature T and the disorder strength (Δ). Our Monte Carlo studies show that the system exhibits 1/f noise related to the electronic avalanche size, which can serve as a model for describing the experimentally observed flicker noise in 2DEG. The power spectrum of our model scales with the frequency with an exponent in the interval 0.3 < α PS < 0.6. We numerically show that the electronic avalanches are scale-invariant with power-law behaviors in and out of the metal-insulator transition line.

The Shcramm-Loewner evolution (SLE) is a correlated exploration process, in which for the chordal set up, the tip of the trace evolves in a self-avoiding manner towards the infinity. The resulting curves are named SLEκ, emphasizing that the process is controlled by one parameter κ which classifies the conformal invariant random curves. This process when experiences some environmental imperfections, or equivalently some scattering random points (which can be absorbing or repelling) results to some other effective scale-invariant curves, which are described by the other effective fractal dimensions and equivalently the other effective diffusivity parameters κ effective . In this paper we use the classical Henon map to generate scattering (absorbing/repelling) points over the lattice in a random way, that realizes the percolation lattice with which the SLE trace interact. We find some meaningful power-law changes of the fractal dimension (and also the effective diffusivity parameter) in terms of the strength of the Henon coupling, namely the z parameter. For this, we have tested the fractal dimension of the curves as well as the left passage probability. Our observations are in support of the fact that this deviation (or equivalently non-zero zs) breaks the conformal symmetry of the curves. Also the effective fractal dimension of the curves vary with the second power of z, i.e. DF (z) − DF (z = 0) ∼ z 2 .

This paper is devoted to the recent advances in self-organized criticality (SOC), and the concepts. The paper contains three parts; in the first part we present some examples of SOC systems, in the second part we add some comments concerning its relation to logarithmic conformal field theory, and in the third part we report on the application of SOC concepts to various systems ranging from cumulus clouds to 2D electron gases.