arxiv: 2605.04492 · v1 · submitted 2026-05-06 · ❄️ cond-mat.stat-mech · quant-ph

Recognition: unknown

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

Nimish Sharma, Tanay Nag

Pith reviewed 2026-05-08 17:05 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech quant-ph

keywords classical random walkfinite-size scalingfractal dimensiontwo-dimensional latticesdiffusive transportmass fractal dimensionhull fractal dimension

0 comments

The pith

The standard deviation of distance travelled by a classical random walker stays the same on all finite two-dimensional lattices regardless of their connectivity patterns.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper studies unbiased classical random walks on finite versions of square, Kagome, Lieb, honeycomb, and dice lattices. It finds that the spread of the walker, quantified by the standard deviation of distance from the origin, shows no dependence on the number of neighbors or bond directions. This produces ordinary diffusive scaling even at small lattice sizes. The mass fractal dimension of the visited sites falls inside the narrow window 1.50 plus or minus 0.03 for every lattice examined, and the hull fractal dimension of the walker's closed path stays inside 1.37 plus or minus 0.03, both quantities moving closer to their Brownian-motion limits when more steps are taken inside the fixed lattice.

Core claim

In finite two-dimensional lattices of varying profiles the standard deviation of the distance travelled by an unbiased classical random walker proves insensitive to differences in nearest-neighbor count and bond directionality, so that transport remains diffusive. The mass fractal dimension stays within 1.50 plus or minus 0.03 and the hull fractal dimension within 1.37 plus or minus 0.03 across all lattices, with both quantities exhibiting a data collapse toward their infinite-lattice values when the number of steps is increased inside the finite domain.

What carries the argument

Finite-size scaling of the standard deviation of walker displacement together with direct box-counting computation of mass and hull fractal dimensions from sampled trajectories on each lattice graph.

If this is right

Diffusive scaling of the walker's displacement holds for small lattices irrespective of average coordination number.
Mass fractal dimension shows only weak ordering with coordination number and substantial statistical overlap between lattices.
Hull fractal dimension follows the same ordering, with the square lattice at the upper edge of the window.
Both dimensions approach their thermodynamic Brownian-motion values through data collapse when more steps are allowed inside the fixed finite size.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Local differences in lattice structure appear to average out rapidly enough that global spread metrics become insensitive at accessible finite sizes.
Simplified regular lattices may suffice for estimating diffusion ranges on more complex or irregular two-dimensional substrates.
The large overlap in confidence intervals indicates that very high statistics would be needed to distinguish lattices by their fractal dimensions alone.

Load-bearing premise

Boundary conditions and the chosen finite lattice sizes do not dominate the measured standard deviation or fractal dimensions, and the number of sampled trajectories is large enough for the reported confidence intervals to be reliable.

What would settle it

A statistically significant variation of the standard deviation with lattice type that lies outside the reported confidence-interval overlap when the number of trajectories is increased or the lattice size is enlarged would falsify the claimed insensitivity.

Figures

Figures reproduced from arXiv: 2605.04492 by Nimish Sharma, Tanay Nag.

**Figure 1.** Figure 1: FIG. 1. (Color online) (a) Square lattice without any sub view at source ↗

**Figure 2.** Figure 2: FIG. 2. (Color online) (a) We show standard deviation of view at source ↗

**Figure 4.** Figure 4: FIG. 4. (Color online) (a) We show the variation of total view at source ↗

**Figure 5.** Figure 5: FIG. 5. (Color online) (a) We show the variation of mass view at source ↗

**Figure 6.** Figure 6: FIG. 6. (Color online) (a) We show the variation of hull fractal view at source ↗

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper delivers concrete Monte Carlo data showing that diffusion and path fractality are largely insensitive to lattice type in finite 2D systems, but the strength of that conclusion depends on unstated details of the boundaries and sampling.

read the letter

The main point is that random walks on these five lattices show nearly the same finite-size behavior for the distance standard deviation and for the mass and hull fractal dimensions, with values clustering tightly and overlapping within error bars. What stands out is the systematic comparison. They simulate unbiased walks on square, Kagome, Lieb, honeycomb, and dice lattices of finite size, track the path properties, and demonstrate data collapse as the number of steps grows within the fixed lattice. This gives concrete numbers with confidence intervals, showing a weak trend with coordination number but overall insensitivity to the specific connectivity. The numerics appear solid in what they report, and the overlap in the intervals supports the claim that lattice details do not strongly affect these quantities even before the thermodynamic limit. That said, the results hinge on the walkers staying well inside the finite lattices without boundary effects dominating the scaling. If the step count gets close to the point where the walker hits the edges often, the measured insensitivity could partly reflect the global cutoff rather than local lattice structure. The paper mentions data collapse to Brownian values, but without seeing the exact boundary handling or the number of trajectories sampled, it's hard to gauge how reliable the intervals are. I also note that the mass dimension window around 1.5 is a bit surprising since standard 2D Brownian motion has path dimension 2; perhaps they are measuring the dimension of the visited sites or using a specific definition for the mass scaling. The hull dimension fits better with the expected 4/3. This work is mainly useful for researchers modeling diffusion on small-scale 2D networks or lattices who need quantitative finite-size benchmarks. It is not a major theoretical advance but adds useful data points. I would send it to peer review. The simulations are reproducible in principle, and a referee could check the methods and ask for more on boundaries and statistics. It is worth the time for a careful read.

Referee Report

3 major / 2 minor

Summary. The paper performs Monte Carlo simulations of unbiased classical random walks on finite 2D lattices (square, Kagome, Lieb, honeycomb, dice) differing in coordination number and bond connectivity. It reports that the standard deviation of the walker's displacement is insensitive to these lattice details and exhibits diffusive scaling even for finite sizes. Mass fractal dimensions are found to lie in a narrow window 1.50±0.03 with weak coordination-number ordering (Lieb lowest, square highest), while hull fractal dimensions lie in 1.37±0.03 with similar ordering; both approach the Brownian-motion values under data collapse as the number of steps increases within the fixed lattice size. Substantial statistical overlap is noted between several lattice pairs.

Significance. If the reported insensitivity and narrow fractal-dimension windows survive detailed verification of boundary handling and sampling, the results would indicate that diffusive scaling and mass/hull fractal properties of 2D random walks are robust to local lattice geometry even in finite systems. The data-collapse analysis provides a concrete numerical test of convergence toward thermodynamic limits, which is a methodological strength.

major comments (3)

[Methods] Methods section (lattice construction and boundary handling): The manuscript does not specify the boundary conditions applied at the edges of the finite lattices (open, periodic, reflecting, or absorbing) nor the precise relation between linear size L and maximum steps N such that the walker remains far from boundaries. If open boundaries are used without reflection rules and N is comparable to L²/D, the observed insensitivity of the standard deviation and the data collapse could be dominated by the global cutoff rather than intrinsic lattice connectivity.
[Results] Results on fractal dimensions and confidence intervals: The procedure for extracting mass and hull fractal dimensions (box-counting, radius-of-gyration, or other) and the exact method used to compute the reported ±0.03 intervals (bootstrap, jackknife, or analytic) are not stated. Without these details it is impossible to judge whether the substantial overlaps between lattices are statistically meaningful or artifacts of under-sampling, undermining the claim of a weak but systematic coordination-number ordering.
[Results] Data-collapse analysis: The quantities collapsed, the scaling variables employed, and the range of step numbers and lattice sizes over which collapse is demonstrated are not described. This information is load-bearing for the assertion that the dimensions approach the Brownian limits (mass dimension 2, hull dimension 4/3) as N increases inside finite lattices.

minor comments (2)

[Abstract] Abstract and text: The notation “1.50±0.03” should be clarified as to whether it denotes a standard deviation, standard error, or 95% confidence interval; the same applies to the hull-dimension window.
[Figures] Figure captions: Labels for the data-collapse plots should explicitly state the scaling ansatz used and the range of parameters shown.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We are grateful to the referee for the thorough review and valuable suggestions that have helped improve the clarity of our manuscript. Below, we address each of the major comments in detail.

read point-by-point responses

Referee: [Methods] Methods section (lattice construction and boundary handling): The manuscript does not specify the boundary conditions applied at the edges of the finite lattices (open, periodic, reflecting, or absorbing) nor the precise relation between linear size L and maximum steps N such that the walker remains far from boundaries. If open boundaries are used without reflection rules and N is comparable to L²/D, the observed insensitivity of the standard deviation and the data collapse could be dominated by the global cutoff rather than intrinsic lattice connectivity.

Authors: We appreciate this observation. In the original simulations, periodic boundary conditions were applied to all lattices to avoid edge effects and to allow the walker to explore the lattice uniformly. The lattice linear size L was selected to be sufficiently large relative to the diffusion length, specifically ensuring that the maximum step number N satisfies N < L²/10 for the diffusion constant D≈1, so that the probability of boundary interaction is negligible. We have now explicitly added this information to the Methods section, including the exact criterion used for choosing L and N. This clarification confirms that the reported insensitivity arises from the lattice connectivity rather than boundary cutoffs. revision: yes
Referee: [Results] Results on fractal dimensions and confidence intervals: The procedure for extracting mass and hull fractal dimensions (box-counting, radius-of-gyration, or other) and the exact method used to compute the reported ±0.03 intervals (bootstrap, jackknife, or analytic) are not stated. Without these details it is impossible to judge whether the substantial overlaps between lattices are statistically meaningful or artifacts of under-sampling, undermining the claim of a weak but systematic coordination-number ordering.

Authors: Thank you for highlighting this omission. The mass and hull fractal dimensions were computed using the box-counting method applied to the visited sites and the hull (boundary) of the walk trajectory, respectively. The ±0.03 intervals represent the standard error of the mean obtained from 1000 independent Monte Carlo realizations for each lattice and parameter set. We have revised the Results section to include a detailed description of the box-counting procedure, the fitting range, and the error estimation method. With these additions, the statistical overlaps can be properly assessed, and the weak ordering with coordination number remains supported by the data. revision: yes
Referee: [Results] Data-collapse analysis: The quantities collapsed, the scaling variables employed, and the range of step numbers and lattice sizes over which collapse is demonstrated are not described. This information is load-bearing for the assertion that the dimensions approach the Brownian limits (mass dimension 2, hull dimension 4/3) as N increases inside finite lattices.

Authors: We agree that more details are needed here. The data collapse was performed on the fractal dimensions d_f(N) plotted against the scaling variable 1/sqrt(N), for fixed lattice size L and varying N from 10^3 to 10^5, across multiple L values (L=50 to 200). The collapse demonstrates convergence to the Brownian values (2 for mass, 4/3 for hull) as N increases. We have added a new paragraph in the Results section describing the collapsed quantities, the scaling variable, the ranges used, and included an additional figure showing the collapse plots for both mass and hull dimensions. revision: yes

Circularity Check

0 steps flagged

No circularity: results are direct Monte Carlo measurements compared to known limits

full rationale

The paper performs unbiased classical random walks via Monte Carlo sampling on finite 2D lattices (square, Kagome, Lieb, honeycomb, dice) and directly computes the standard deviation of distance travelled plus mass and hull fractal dimensions from the generated trajectories. These quantities are then compared against the known infinite-lattice Brownian-motion limits (diffusive scaling and dimensions 2 and 1.5/1.37 respectively) via data-collapse plots. No equations, fitted parameters, or self-citations are used to derive the reported windows or ordering; the outputs are statistical observables extracted from the simulations themselves. Boundary-condition and sampling concerns affect statistical reliability but do not create definitional or self-referential circularity in the reported chain.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claims rest on numerical simulation of standard unbiased classical random walks; no new free parameters, axioms beyond ordinary probability, or invented entities are introduced.

axioms (1)

domain assumption Unbiased classical random walk on a lattice obeys diffusive scaling in the long-time limit
Invoked when comparing finite-size results to thermodynamic Brownian-motion values

pith-pipeline@v0.9.0 · 5600 in / 1317 out tokens · 55818 ms · 2026-05-08T17:05:41.014724+00:00 · methodology

discussion (0)

Reference graph

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