Correlation effects in the diluteness pattern in non-integral dimensional systems on $\nu=\frac{4}{5}$ superdiffusion process

TL;DR

This paper investigates how correlations in non-integer dimensional systems influence superdiffusive behavior, using loop erased random walks and Ising model correlations, revealing critical exponents and fractal properties at phase transition.

Contribution

It introduces a novel analysis of correlation effects on superdiffusion in non-integer dimensions using Ising-based models and identifies critical geometrical and scaling properties.

Findings

At critical temperature, the superdiffusion exponent is approximately 0.807.

The fractal dimension scales inversely with the square root of the correlation length.

Trace properties are consistent with Schramm-Loewner evolution theory.

Abstract

The effect of the correlations in the diluteness pattern in the systems with non-integral dimensionality, on $\nu=\frac{4}{5}$ superdiffusion process is considered in this paper. These spatial correlations have proved to be very effective in the critical phenomena. To simulate the particles motion in this process, we employ the loop erased random walk (LERW). The spatial correlations between imperfections (site-diluteness) also have been modeled by the Ising model on a square lattice. It models the forbidden regions into which the particles are not allowed to enter. The correlations are controlled by an artificial temperature $T$. The trace of the walkers is shown to be self-similar, whose fingerprint is the power-law behaviors. The detailed analysis of the random walker's traces reveal that the (Ising-type) correlations affect their geometrical properties. At the critical artificial temperature $T_c$ we observe that the exponent of end-to-end distance $\nu$ becomes $0.807\pm 0.002$. The fractal dimension of the walker's trace is the other geometrical quantity which scales inversely with the square root of the correlation length of the Ising model, i.e. $D_f(T)-D_f(T_c)\sim \zeta^{-\alpha}$ in which $\alpha=0.43\pm 0.05$. The winding angle test also reveals that the traces are compatible with the Schramm-Loewner evolution theory, and the diffusivity parameter for $T=T_c$ is $\kappa=1.89\pm 0.05$.