Scaling Analysis of Random Walks with Persistence Lengths: Application to Self-Avoiding Walks

Abstract

We develop an approach for performing scaling analysis of $N$-step Random Walks (RWs). The mean square end-to-end distance, $⟨{\vec{R}}_N^2⟩$, 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 $j$-th step. For RW models statistically invariant under orthogonal transformations, we analytically introduce a relation between $⟨{\vec{R}}_N^2⟩$ 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 $⟨{\vec{R}}_N^2⟩$. Building SAWs with typically one hundred steps, we estimate the exponents ${\nu }_0$ and ${\Delta }_1$ 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 $⟨{\vec{R}}_N^2⟩$, being that the whole information needed is contained in the inner part of the paths.

arXiv comment12 pages, 3 figures

References in corpus (13)

• Accurate estimate of the critical exponent $\nu$ for self-avoiding walks via a fast implementation of the pivot algorithm
• Enumeration of self-avoiding walks on the square lattice
• The Random Transverse Ising Spin Chain and Random Walks
• A review of Monte Carlo simulations of polymers with PERM
• Monte Carlo Methods for the Self-Avoiding Walk
• Exact enumeration of self-avoiding walks
• Correction-to-scaling exponents for two-dimensional self-avoiding walks
• Efficient implementation of the pivot algorithm for self-avoiding walks
• Analytical Results for the Statistical Distribution Related to Memoryless Deterministic Tourist Walk: Dimensionality Effect and Mean Field Models
• The persistence length of two dimensional self avoiding random walks
• Universal monomer dynamics of a two dimensional semi-flexible chain
• Asymptotic scaling behavior of self-avoiding walks on critical percolation clusters
• Distance statistics in random media: high dimension and/or high neighborhood order cases

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