Self-Avoiding Walk on the square site-diluted Ising-correlated lattice

TL;DR

This study investigates the behavior of self-avoiding walks on a correlated percolation lattice influenced by the Ising model, confirming Flory's relation at criticality and revealing new fractal dimension behavior off-criticality.

Contribution

It introduces an enriched Monte Carlo approach and analyzes the fractal dimension dependence on Ising correlation length, extending understanding of self-avoiding walks in correlated disordered systems.

Findings

Exponents agree with Flory's approximation at critical Ising system.

Discovered a new relation for fractal dimension off-critical, proportional to 1/√ξ(T).

Extracted SLE parameter κ indicating conformal invariance.

Abstract

The self-avoiding walk on the square site-diluted correlated percolation lattice is considered. The Ising model is employed to realize the spatial correlations of the metric space. As a well-accepted result, the (generalized) Flory's mean field relation is tested to measure the effect of correlation. After exploring a perturbative Fokker-Planck-like equation, we apply an enriched Rosenbluth Monte Carlo method to study the problem. To be more precise, the winding angel analysis is also performed from which the diffusivity parameter of Schramm-Loewner evolution (SLE) theory ($\kappa$) is extracted. We find that at the critical Ising (host) system the exponents are in agreement with the Flory's approximation. For the off-critical Ising system we find also a new behavior for the fractal dimension of the walker trace in terms of the correlation length of the Ising system $\xi(T)$, i.e. $D_F^{\text{SAW}}(T)-D_F^{\text{SAW}}(T_c)\sim \frac{1}{\sqrt{\xi(T)}}$.