Finite-size scaling properties of classical random walk on various two-dimensional lattices

TL;DR

This study investigates how different two-dimensional lattice structures affect classical random walk properties, revealing insensitivity of diffusion to lattice non-uniformity and analyzing fractal dimensions and scaling behaviors.

Contribution

It provides a comparative analysis of mass and hull fractal dimensions across various finite-size 2D lattices, highlighting statistical overlaps and scaling trends.

Findings

Standard deviation of walk distance is insensitive to lattice non-uniformity.

Mass fractal dimension varies within 1.50 ± 0.03 across lattices.

Hull fractal dimension is within 1.37 ± 0.03, with square lattice showing the upper bound.

Abstract

We consider various two-dimensional lattices such as square, Kagome, Lieb, honeycomb, dice lattices of finite extent, to study the effect of lattice profile in terms of the number of nearest neighbour and connectivity patterns on the classical random walk in the unbiased scenario. We find that the standard deviation of distance travelled by the walker is insensitive to the non-uniformity of the lattice profile leading to diffusive transport even in the finite size lattices. Our study indicates that the mass fractal dimension varies within a window $1.50\pm 0.03$ for all finite-size lattices. A weak ordering within the above window, correlated with the average coordination number, is observed, while Lieb and square lattices yielding the minimum and maximum values, respectively. However, confidence intervals reveal substantial statistical overlap for several lattice pairs even though the lattice profiles vary as far as the average number of connecting bonds and directionality of bonds are concerned. We also study the scaling complexity of the circumference of the closed curve traced by the walker while investigating the hull dimension. We find similar trend for hull fractal dimension as well and that was found to within the window $1.37\pm 0.03$ for finite-size lattices. Within the above window, the ordering remains qualitatively unaltered as compared to mass dimension while the confidence interval rectifies the order quantitatively. The square lattice clearly exhibits the upper bound for hull fractal dimension and the remaining lattices show extensive statistical overlap within the above window. We exhibit a tendency of the mass and hull fractal dimension to reach their thermodynamic values given by Brownian motion when we allow more number of steps within the finite size of the lattice, as confirmed by a data collapse analysis.