Scaling analysis of random walks with persistence lengths: Application to self-avoiding walks

We develop an approach for performing scaling analysis of $N$-step random walks (RWs). The mean square end-to-end distance, $\langle {\vec{R}}_N^2\rangle$, is written in terms of inner persistence lengths (IPLs), which we define by the ensemble averages of dot products between the walker's position and displacement vectors, at the $j\mathrm{th}$ step. For RW models statistically invariant under orthogonal transformations, we analytically introduce a relation between $\langle {\vec{R}}_N^2\rangle$ and the persistence length, ${\lambda }_N$, which is defined as the mean end-to-end vector projection in the first step direction. For self-avoiding walks (SAWs) on 2D and 3D lattices we introduce a series expansion for ${\lambda }_N$, and by Monte Carlo simulations we find that ${\lambda }_{\infty }$ is equal to a constant; the scaling corrections for ${\lambda }_N$ can be second- and higher-order corrections to scaling for $\langle {\vec{R}}_N^2\rangle$. Building SAWs with typically 100 steps, we estimate the exponents ${\nu }_0$ and ${\mathrm{\Delta }}_1$ from the IPL behavior as function of $j$. The obtained results are in excellent agreement with those in the literature. This shows that only an ensemble of paths with the same length is sufficient for determining the scaling behavior of $\langle {\vec{R}}_N^2\rangle$, being that the whole information needed is contained in the inner part of the paths.

Figure 1

SAW persistence length for (a) square and (b) cubic lattices. In both lattices, ${\lambda }_N$ converges to a constant. The inset plot (a) shows ${\lambda }_N$ fitted by the function ${\alpha }_0+{\alpha }_1{N}^{-0.34}$ of Ref. [30], where ${\lambda }_{\infty }=2.664\left(3\right)$ is compatible with our estimate ${\lambda }_{\infty }=2.525\left(4\right)$ from Eq. (4). For the cubic lattice, ${\lambda }_{\infty }\sim \sqrt{2}$ is compatible with the one of Ref. [27]. The inset plot (b) depicts the random pattern of the residual plot for ${\lambda }_N$ when fitted by Eq. (4).

Figure 2

IPL data collapse in $log-log$ scale for (a) square and (b) cubic lattices with ($▴$ $N=30$), ($⧫$ $N=40$), ($■$ $N=50$), and ($●$ $N=60$). The ${\langle {\vec{R}}_j·{\vec{u}}_j\rangle }_N$ behaves as linear increasing function up to $\sim j_{\text{max}}$, with a slope $\approx (2{\nu }_0-1)$. In both lattices $j_{\text{max}}\propto N$, with constant of proportionality close to each other ($\sim 0.7$). For $j>j_{\text{max}}$ the scalar products contribute to residual terms of corrections to scaling of $\langle {\vec{R}}_N^2\rangle$.

Figure 3

IPL differences ($\mathrm{\Delta }{R}_j(N_1,N_2)$) for SAWs: (a) for square and (b) cubic lattices, with $N_1=40$ and $N_2=60$, and $N_1=60$ and $N_2=90$, respectively. According to $\mathrm{\Delta }{R}_j(N_1,N_2)$ depicted here [see Eq. (5)], the ${\mathcal{I}}_j$ starts to be influenced for $j_c>N_1/2$ and $j_c>N_1/3$, for 2D and 3D square lattices, respectively. Inset plots show IPL nonweighted fit using Eq. (6), within a confidence interval of $95\%$. The square lattice data includes $N$ ranging from 90 to 120 with increment of 10. The fitting parameters obtained are $\phi =0.4979\left(21\right)$ and ${\beta }_1=0.6630\left(50\right)$. The cubic lattice data includes $N$ ranging from 150 to 195 with increment of 15. The fitting parameters obtained are $\phi =0.1752\left(14\right), {\mathrm{\Delta }}_1=0.522\left(52\right)$, and ${\beta }_1=0.7581\left(56\right)$.