Scaling analysis of Self-Avoiding Walk with Persistence Lengths

The self-avoiding walk (SAW) in regular lattices is a random walk (RW) model, where the walker steps to nearest-neighbor sites and does not visit a site more than once. Due to non-overlapping paths, the SAW plays a central role on Polymer Physics by capturing the excluded volume effect.
A key aspect to study the SAW is the scaling analysis of the conformational quantities. One of these quantities is the average projection of the end-to-end distance, $\vec{R}_{N}=\vec{u}_{1}+\cdots+\vec{u}_{N}$, along the first step direction, $\vec{u}_{1}$, or persistence length $\langle \lambda_{N} \rangle_{N}=\langle \vec{R}_{N}\cdot\vec{u}_{1}\rangle_{N}$, where $|\vec{u}_{1}|=1$. The known scaling predictions for $\langle \lambda_{N}\rangle_{N}$ not even agree one with each others. To address such issue, we write the conformational quantities in terms of the scalar product between the position $\vec{R}_{j}$ and displacement $\vec{u}_{j}$, at the $j$-th step: $\langle\vec{R}_{j}\cdot\vec{u}_{j}\rangle_{N}$, of an $N$-step SAW. The mean square end-to-end distance, $\langle \vec{R}_{N}^{2}\rangle_{N}$, is proportional to the summation of the inner persistence length, $\langle \vec{R}_{j}\cdot\vec{u}_{j}\rangle_{N}$, for $1<j<N$. For the SAW model, we obtain $\langle \lambda_{N}\rangle_{N}=\langle \vec{R}_{N}\cdot\vec{u}_{N}\rangle_{N}$ implying the novel relation $\langle\vec{R}_{N}^{2}\rangle_{N} = \langle \vec{R}_{N-1}^{2}\rangle_{N} + 2\langle \lambda_{N}\rangle_{N} - \vec{u}_{N}^{2}$. Based on the accepted $\langle \vec{R}_{N}^{2}\rangle_{N}\sim N^{2\nu_{0}}$ scaling behavior and Monte Carlo simulations, we find $\langle \lambda_{N}\rangle_{N}$ convergence to a constant value with corrections to scaling, in square and cubic lattices. From Monte Carlo data we find that $\langle\vec{R}_{j}\cdot\vec{u}_{j}\rangle_{N}\approx j^{2\nu_{0}-1}$ for $1<j<j_{max}$, and reaches a maximum value at the $j_{max}$-th step. For $j>j_{max}$, the inner persistence length is no longer increasing, but a monotonic decreasing function that contributes largely with the corrections to scaling of $\langle \vec{R}_{N}^{2}\rangle_{N}$. Such a scaling behavior of inner persistence length is observed in both, square and cubic lattices. We define $\Delta R_{j} = \langle \vec{R}_{j}\cdot\vec{u}_{j}\rangle_{N_{2}}-\langle \vec{R}_{j}\cdot\vec{u}_{j}\rangle_{N_{1}}$, with $N_{2}>N_{1}$, in order to find the step $j_{c}(N)<j_{max}$, where the inner persistence length starts to be notably influenced by the walk length. Finally, considering $1<j<j_{c}(N)$ we obtain an accurate estimate of $\nu_{0}$ from $\langle\vec{R}_{j}\cdot\vec{u}_{j}\rangle_{N}$ scaling relation, for walks with few steps, say $N<100$, in the square and cubic lattice.