https://doi.org/10.1140/epjb/s10051-026-01169-4
Research - Statistical and Nonlinear Physics
Finite-size scaling properties of classical random walk on various two-dimensional lattices
1
Department of Physics, BITS Pilani-Pilani Campus, 333031, Pilani, Rajasthan, India
2
Department of Physics, BITS Pilani-Hyderabad Campus, 500078, Hyderabad, Telangana, India
a
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Received:
22
October
2025
Accepted:
10
April
2026
Published online:
5
May
2026
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 i.e., root mean square displacement of the walker is insensitive to the non-uniformity of the lattice profile leading to diffusive transport even in the finite size lattices. We next study the scaling complexity of the entire closed curve traced by the walker while investigating the bulk and boundary fractal dimension namely mass and hull dimensions. Our study indicates that the mass fractal dimension varies within a window 
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 a similar trend for hull fractal dimension as well and that was found to within the window 
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. Therefore, our study uncovers the finite-size effect of lattice geometry and co-ordination number on the scaling properties of the path of the random walker.
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© The Author(s), under exclusive licence to EDP Sciences, SIF and Springer-Verlag GmbH Germany, part of Springer Nature 2026
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
