Invasion percolation in short-range and long-range disorder background

TL;DR

This study explores invasion percolation in correlated disordered backgrounds, revealing distinct behaviors between short-range and long-range interactions, including a new dynamical crossover in long-range cases.

Contribution

It introduces a comparative analysis of invasion percolation in short-range and long-range correlated disorder backgrounds, highlighting the robustness of SRI properties and the unique long-range effects.

Findings

Fractal dimension of RCP IP external frontier is approximately 1.099.

SRI IP exhibits a fractal dimension close to 4/3, similar to normal IP.

A novel dynamical crossover in RCP IP divides behavior into three regimes.

Abstract

In this paper, we investigate the invasion percolation (IP) in imperfect support in which the configuration of imperfections is considered to be correlated. Three lattice models were engaged to realize this pattern: site percolation, Ising model and random Coulomb potential (RCP). The first two models are short range interaction (SRI), whereas the last one includes coulomb like interactions which is pretty long range (long-range interactions, LRI). By examining various dynamical observables we show that the critical exponents of SRI IP are robust against the control parameters (temperature in the Ising model and occupation probability in site percolation), whereas its properties in the LRI (RCP) supports are completely different from the normal IP (i.e. on the regular lattice). Especially the fractal dimension of the external frontier of the largest hole converges to $1.099\pm 0.008$ for RCP IP, whereas it is nearly $\frac{4}{3}$ for SRI IP being compatible with normal IP. Additionally a novel dynamical crossover is seen in the RCP IP according to which the time dependence of all of the observables is divided to three parts: the power-law (small times), the logarithmic (mid time), and the linear (long time) regimes. The second crossover time is shown to go to infinity in the thermodynamic limit, whereas the first crossover time is nearly unchanged, signaling the dominance of the logarithmic regime. The observables become nearly constant in the thermodynamic limit for the long time, showing that it is a stationary phase.