Self-Attracting Walk on Lattices

TL;DR

This paper investigates a self-attracting walk model on lattices using Monte Carlo simulations, revealing diffusive behavior in one dimension and non-universal exponents in higher dimensions depending on attraction strength.

Contribution

It provides a detailed analysis of the self-attracting walk's scaling behavior across different dimensions and interaction strengths, highlighting non-universal critical exponents.

Findings

Diffusive behavior in 1D with exponents 1/2

Non-universal exponents in 2D and 3D

Exponent variation depends on attraction strength

Abstract

We have studied a model of self-attracting walk proposed by Sapozhnikov using Monte Carlo method. The mean square displacement $ < R^2(t) > \sim t^{2\nu}$ and the mean number of visited sites $ < S(t) > \sim t^{k}$ are calculated for one-, two- and three-dimensional lattice. In one dimension, the walk shows diffusive behaviour with $\nu=k=1/2$. However, in two and three dimension, we observed a non-universal behaviour, i.e., the exponent $\nu$ varies continuously with the strength of the attracting interaction.