Self-avoiding walk on a square lattice with correlated vacancies

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 angle analysis is also performed from which the diffusivity parameter of Schramm-Loewner evolution theory ($\kappa$) is extracted. We find that at the critical Ising (host) system, the exponents are in agreement with Flory's approximation. For the off-critical Ising system, we find also a behavior for the fractal dimension of the walker trace in terms of the correlation length of the Ising system $\xi \left(T\right)$, i.e., $D_F^{\text{SAW}}\left(T\right)-D_F^{\text{SAW}}\left(T_c\right)\sim \frac{1}{\sqrt{\xi \left(T\right)}}$.

Figure 1

Three situations of SAWs reaching the point $\vec{r}$ at time $t$.

Figure 2

$N=2000$ bulk SAW sample in an Ising sample media in a $512\times 512$ lattice at $T=T_c$ (blue lines). The red sites represent the forbidden (inactive) sites, and the white sites are representative of the active ones.

Figure 3

(a) ${log}_{10}{\langle R^2\rangle }^{\frac{1}{2}}$ in terms of ${log}_{10}N$ for $T=T_c$. Inset: ${log}_{10}〈N\left(L\right)〉$ in terms of ${log}_{10}L$. (b) $〈{\theta }^2〉$ in terms of $lnN$ with the slope $\gamma =1.77\pm 0.02$ for $T=T_c$. Upper inset: the distribution function of $\theta$ for $N=200$. Lower inset: $〈\theta 〉$ in terms of $N$, which is zero.

Figure 4

(a) $ln〈N\left(L\right)〉$ in terms of $lnL$ for various rates of temperature. Upper inset: $D_F$ in terms of $T$; lower inset: power-law behavior of the fractal dimension. (b) $ln{\langle R^2\rangle }^{\frac{1}{2}}$ in terms of $lnN$ for various rates of temperature. Upper inset: the $\nu$ exponent in terms of $T$; lower inset: power-law behavior of $\nu$. (c) The distribution of the winding angle $\theta$ for $N=200$ and $T=1.768$. (d) $〈{\theta }^2〉$ in terms of $lnN$ with the slope $\gamma$ which is $T$-dependent. Upper inset: $\gamma$ in terms of $T$; lower inset: power-law behavior of $\gamma$.