Surprising variants of Cauchy's formula for mean chord length

Diana Thongjaomayum; Prabodh Shukla

REVIEW 4 major objections 4 minor 12 references

Surprising variants of Cauchy's formula for mean chord length

T0 review · 4 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A random walk from boundary to boundary of an $N^d$ hypercubic lattice has mean length $N$ and a $n^{-3/2}$ power-law tail.

desk verdict A clean lattice version of Cauchy's mean chord formula with a heuristic—not proven—power-law tail in 2D/3D. read the letter →

arxiv 1908.06608 v2 pith:2OBEB4ZZ submitted 2019-08-19 cond-mat.stat-mech

Prabodh Shukla , Diana Thongjaomayum This is my paper · ORCID

classification cond-mat.stat-mech

keywords Cauchy'sformulameanchordlengthrandomwalkhypercubiclatticegambler'sruinpower-lawdistributionfirst-returntimefirst-passage

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper establishes a lattice analogue of Cauchy's mean-chord formula: a random walk on an $N \times N$ square or $N \times N \times N$ cubic lattice that starts at a uniformly chosen boundary site and stops at its first return to the boundary has mean length $\langle n \rangle = N$, the volume-to-surface ratio in lattice units. The same linear scaling holds for certain anisotropic walks after fixed rescaling. The paper also reports that for symmetric walks the distribution of lengths decays as $P(n) \sim n^{-3/2}$ for large $n$, a power law rather than the exponential distribution of continuum mean-field treatments. This matters because it shows that the mean chord length depends only on $V/S$ and not on the detailed zig-zag path inside the body, and it offers a simple explanation of Cauchy's formula through the one-dimensional gambler's ruin problem.

What carries the argument

The mechanism is the projection of the lattice walk onto one coordinate axis, which is a one-dimensional random walk between two absorbing endpoints, the gambler's ruin problem. The exact solution gives the mean absorption time $m(N-m)$ from site $m$, hence $N$ for a boundary-to-boundary walk, and a hitting-time distribution decaying as $n^{-3/2}$. The paper's argument then treats the $d$-dimensional walk as a set of independent such components along each axis, so the linear mean and the qualitative power-law tail transfer from one dimension to two and three dimensions; the same one-dimensional formula explains the anisotropic cases, including the weight factors that collapse the mean-length data.

What would settle it

Compute the exact first-return-time distribution of the isotropic two-dimensional walk to the square boundary by transfer-matrix or eigenvalue methods, or by very high-statistics simulation at $N=10^4$, and measure the log-log slope in the window $N \ll n \ll N^2$; a slope clearly different from $-3/2$, or a slope that shifts with $N$, would refute the central claim. A direct check of the component-independence assumption is to measure whether the tail distribution depends on the exit coordinate along one axis: if it does, the reduction to independent components fails.

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Extended reading notes

Core claim

The central discovery is that Cauchy's formula survives when straight chords are replaced by restricted lattice random walks: an isotropic walk that enters an $N^d$ hypercubic lattice through a uniformly chosen surface site and leaves at the first surface hit has mean length equal to $N$, independently of $d$ in the dimensions studied. The distribution of walk lengths is qualitatively different from the continuum mean-field exponential: it decays as $C n^{-3/2}$ over many decades for a symmetric walk. For anisotropic walks the mean still scales linearly with $N$ after multiplying by fixed weight factors, and the power-law tail persists precisely when the walk is symmetric between each pair of opposite faces; breaking that symmetry replaces the tail with a finite-size peak. The paper argues that all of these features reduce to the one-dimensional two-absorber random walk.

Load-bearing premise

The load-bearing premise is that the components of the walk along the coordinate axes are independent, so the one-dimensional $n^{-3/2}$ tail transfers unchanged to two and three dimensions, and that 'large $n$' means the intermediate window $N \ll n \ll N^2$ before the one-dimensional finite-size exponential cutoff sets in.

Editorial extensions

If this is right

If the main claim is correct, the mean first-return length on an $N^d$ hypercubic lattice is $N$ for every dimension studied, so the volume-to-surface ratio controls the mean even when a single step does not resemble a chord.
The $n^{-3/2}$ tail implies that long walks are abundant: the probability of a walk lasting $n$ steps decays only polynomially, so fluctuations around the mean are much larger than an exponential model would allow.
The lattice derivation ties Cauchy's formula to the gambler's ruin problem, giving a simple route to the chord-length formula that does not require the Boltzmann transport equation.
When the walk is symmetric between opposite faces, the power-law tail holds; when that symmetry is broken, the same mean scaling survives but the power-law tail is replaced by a finite-size peak.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Editorial inference: the component-independence assumption suggests a testable factorization of the $d$-dimensional first-return-time distribution into one-dimensional factors in the window $N \ll n \ll N^2$; measuring correlations between exit times and exit coordinates would confirm or refute it.
Editorial inference: for rectangular or other non-cubic domains, the mean walk length should still scale with the volume-to-surface ratio, but the tail exponent and the location of the finite-size cutoff may depend on aspect ratio.
Editorial inference: the loopy, localized structure of surface-to-surface walks resembles a bounded-domain model of anomalous diffusion, where the power-law tail could be observed as a slowly decaying return-time distribution in single-particle-tracking experiments.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Dear Colleague,

The paper's 1D analytic core is classical (Ellis/Feller), and for symmetric surface-to-surface walks on hypercubic lattices the mean-length result =N is well supported by analytics and simulations. That part is a clean, if modest, lattice rendering of Cauchy's formula, and the paper gives proper credit to the continuum work in refs [7,8]. What is genuinely new is the claim that the walk-length distribution in d=2,3 decays as n^{-3/2}, and that is where the paper gets soft.

The component-independence step after Eq. (2) is an assertion, not a proof. In discrete time, each step changes exactly one coordinate, so the coordinate hitting times are dependent; the exit time is the minimum of those hitting times. Nothing in the paper shows the dependence preserves the one-dimensional tail. On top of that, for fixed N the state space is finite, so the exact tail is exponential for n >> N^2; the power-law can only hold in an intermediate window. The paper says "for large n" without specifying this, which is at best misleading. The abstract also states "mean length is equal to N" without the symmetric-walk caveat, though the body is more careful. The weighting factors 0.89 and 1.20 used to collapse the asymmetric data are fitted, not derived—a minor point compared to the tail gap.

These gaps are addressable. The authors should state a precise intermediate-asymptotic result and either derive the d=2,3 tail or provide a controlled approximation. As it stands, the paper is a physically motivated heuristic with clean simulations and a solid 1D anchor, not a rigorous theorem. It deserves referee time—a serious referee could push the authors to make the window explicit and test the component-independence assumption directly—but the decision should be conditional on that revision.

For a reader, it is a useful note if you care about lattice first-passage or stereology. I would put it in the "maybe" pile for a reading group, and I would not cite the tail claim in its current form. I would accept it for peer review, with the expectation of major revisions.

Best

Referee Report

4 major / 4 minor

Summary. The paper studies nearest-neighbor random walks on d-dimensional hypercubic lattices (d = 1, 2, 3) that start on the boundary and terminate upon first return to the boundary. It reports two main claims: (i) the mean walk length <n> is proportional to N, with <n> = N for symmetric walks; and (ii) the length distribution P(n) decays as n^{-3/2} for large n for symmetric walks and for some asymmetric cases. These claims are presented as a lattice analogue of Cauchy's chord-length formula. The one-dimensional case is treated with classical gambler's-ruin formulas (Eqs. (1)-(2)); the higher-dimensional results are supported by simulations with N up to 10^3 and by a heuristic component-independence argument.

Significance. If the n^{-3/2} tail in d = 2, 3 were rigorously established, the paper would give a simple and appealing discrete counterpart of recent continuum results on Cauchy's formula, with potential applications in first-passage and transport problems. The simulations are extensive (more than 10^6 realizations for N = 10^3), and the one-dimensional exact formulas are a useful starting point. However, the higher-dimensional distributional claim is the principal novelty and is currently supported only by an unproved heuristic; the manuscript also overstates the mean-length claim. The significance is therefore conditional on a rigorous or clearly delimited version of the distributional result.

major comments (4)

[Text following Eq. (2)] The step 'On a d-dimensional hypercubic lattice we may consider components of the walk along each dimension independently' is not a proof. In discrete time the coordinate processes are dependent, because each step changes exactly one coordinate, and the exit time is the minimum of the dependent coordinate hitting times. No error estimate or argument is given to show that this dependence preserves the one-dimensional n^{-3/2} tail in d = 2, 3. Since this tail is the paper's novel claim, the manuscript should either supply a rigorous intermediate-asymptotic derivation or explicitly present it as a conjecture supported by simulation.
[Abstract and Fig. 3 discussion] The statement that P(n) scales as n^{-1.5} 'for large n' is ambiguous and, as a fixed-N asymptotic statement, false. Equation (2) contains factors cos^{n-1}(nu pi / N), so for fixed N and n >> N^2 the distribution decays exponentially, not as a power law. The power law can hold only in the intermediate window 1 << n << N^2. The paper should state this window explicitly and verify that the data in Figs. 3 and 4 are collected within it; without that qualification, the claim is literally incorrect as a fixed-N asymptotic statement.
[Abstract and conclusion] The abstract and the concluding paragraph claim without qualification that 'the mean length of walks is equal to N'. The body shows this is not true for the anisotropic cases: the two two-dimensional asymmetric datasets are multiplied by fitted factors 0.89 and 1.20 to collapse onto the <n> = N line, and in one dimension the asymmetric formula (1) gives proportionality constants that depend on the bias and the starting distribution. The abstract should carry the same caveat as the body: <n> is proportional to N, with <n> = N only for symmetric walks in the large-N limit and under the specified boundary-entry convention.
[Eq. (1)] Equation (1) contains an off-by-one error. For a simple symmetric random walk on sites 1, ..., N with absorbing endpoints 1 and N, the expected duration from site m is (m-1)(N-m), not m(N-m); for N = 3 and m = 2, the formula gives S_2 = 2 while the true value is 1. This error propagates into the claimed proportionality constants and into the empirical 0.89 and 1.20 rescaling factors used in Fig. 2, so Eq. (1) and the statements depending on it must be corrected before the quantitative claims can be accepted.

minor comments (4)

[Title and affiliations] There are several typographical errors: 'leng th' in the title, 'Shil long' for Shillong, and 'Euvres' should be 'Oeuvres' in the first reference.
[Fig. 3 caption] The caption refers to 'open black squares', but the legend lists line styles and colors; please clarify which dataset corresponds to the non-power-law case.
[Fig. 4 discussion] The text says 'The simulation data agrees with the analytic solution mentioned above but the peak is a finite size effect', yet earlier it says the analytic result applies to n <= N while the peak occurs at n about 2N. This needs clarification of which regimes are covered by the analytic formula.
[Eq. (2)] The rendering of Eq. (2) is ambiguous: '2np' should presumably read '2^n p' or similar. Please correct the typography so the factors are unambiguous.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the 1D mean and tail are standard external results, and the higher-dimensional power law is an asserted extrapolation rather than a fitted or definitionally forced prediction.

full rationale

The paper's only analytical derivation is the one-dimensional gambler's ruin result quoted from Feller/Ellis (Eqs. 1-2); from this it extracts <n> linear in N and P(n) ~ n^-3/2 in 1D. The 2D and 3D claims are then made by the heuristic that coordinate components may be considered independently. This is an unproved independence assumption and a possible correctness gap, but it is not circular: the higher-dimensional distribution is not equal by construction to the 1D formula, nor is any fitted parameter substituted for the claimed exponent. The rescaling factors 0.89 and 1.20 used to collapse the asymmetric <n> data are empirical weights, but they are ancillary, are not asserted to be derived from step probabilities, and do not enter the symmetric mean or power-law claims. There are no self-citations used as load-bearing premises; the references to prior random-walk/Cauchy work ([7], [8]) are to continuum results used as motivation and boundary-condition analogies, not to the authors' own unverified results. Hence the derivation chain is not circular, even though the 2D/3D tail exponent is under-supported.

Assumptions & free parameters 1 free parameters · 3 assumptions · 0 invented entities

The central mean-length claim for symmetric walks rests on classical 1D gambler's ruin results and a uniform-start/inward-first-step condition borrowed from continuum Cauchy formula studies. The higher-dimensional power-law claim additionally rests on an unproved component-independence heuristic. The only fitted numbers are two weighting factors used to collapse asymmetric data.

free parameters (1)

Asymmetric case weighting factors = 0.89 and 1.20
Chosen to collapse the two asymmetric 2D <n> versus N datasets onto the symmetric line in Fig.2; not derived from the stated step probabilities (0.10, 0.10, 0.40, 0.40) and (0.10, 0.30, 0.30, 0.30).

assumptions (3)

standard math The 1D gambler's ruin formulas (Eqs. 1 and 2) for mean and distribution of absorption times are correct and applicable to the lattice walk along a coordinate.
Invoked as the analytic anchor for the mean <n>=N and the n^-3/2 tail in one dimension.
domain assumption Starting points uniformly distributed over the boundary with the first step directed inward is the lattice analogue of the uniform isotropic incident flux required by Cauchy's formula.
Used to connect lattice walk averages to Cauchy's chord formula, following the condition stated in ref [8].
ad hoc to paper In d dimensions the walk's exit statistics can be understood by considering its components along each coordinate independently (with rescaled time), and no proof or error bound is supplied.
This is the load-bearing heuristic that transfers the 1D power-law tail to d=2,3; it is asserted rather than derived.

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Cite this review

Pith. "Pith review of Surprising variants of Cauchy's formula for mean chord length." pith.science (2026). https://pith.science/paper/2OBEB4ZZ

@misc{pith2026190806608,
  author       = {Pith},
  title        = {Pith review of: Surprising variants of Cauchy's formula for mean chord length},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2OBEB4ZZ}},
  note         = {Machine review of arXiv:1908.06608}
}

abstract

We examine isotropic and anisotropic random walks which begin on the surface of linear ($N$), square ($N \times N$), or cubic ($N \times N \times N$) lattices and end upon encountering the surface again. The mean length of walks is equal to $N$ and the distribution of lengths $n$ generally scales as $n^{-1.5}$ for large $n$. Our results are interesting in the context of an old formula due to Cauchy that the mean length of a chord though a convex body of volume $V$ and surface $S$ is proportional to $V/S$. It has been realized in recent years that Cauchy's formula holds surprisingly even if chords are replaced by irregular insect paths or trajectories of colliding gas molecules. The random walk on a lattice offers a simple and transparent understanding of this result in comparison to other formulations based on Boltzmann's transport equation in continuum.

Figures

Figures reproduced from arXiv: 1908.06608 by the authors.

**Figure 1.** FIG. 1: An isotropic random walk on a 10 [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗

**Figure 2.** FIG. 2: Average length [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

Figures from the paper (2 more)

**Figure 3.** Figure 3: FIG. 3: Probability of a walk comprising [PITH_FULL_IMAGE:figures/full_fig_p006_3.png]

**Figure 4.** Figure 4: FIG. 4: Similar plot as in Fig.3 but without logarithmic binn [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages

[1]

Cauchy, Euvres compl` etes: Series 1 (Cambridge Library Collection-Mathematics), Cam bridge University Press (2009)

A. Cauchy, Euvres compl` etes: Series 1 (Cambridge Library Collection-Mathematics), Cam bridge University Press (2009). doi:10.1017/CB09780511702389

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F. A. Oliveira, R. M. S. Ferreira, L. C. Lapas, and M. H. Va instein, Front. Phys. 7:18 (2019); doi: 10.3389/fphy.2019 .00018 5 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 y x FIG. 1: An isotropic random walk on a 10 × 10 lattice that starts and ends on the boundary

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[12]

M. R. Evans and S. N. Majumdar, Phys. Rev. Lett. 106, 1606 01 (2011). 6 0 250 500 750 1000 0 250 500 750 1000 <n> N <n>=N d=1: p=0.50;0.50 d=1: p=0.75;0.25 d=2: p=0.25;0.25;0.25;0.25 d=2: p=0.10;0.10;0.40;0.40 d=2: p=0.10;0.30;0.30;0.30 d=3: p=1/6;1/6;1/6;1/6;1/6;1/6 FIG. 2: A...

2011

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Cauchy, Euvres compl` etes: Series 1 (Cambridge Library Collection-Mathematics), Cam bridge University Press (2009)

A. Cauchy, Euvres compl` etes: Series 1 (Cambridge Library Collection-Mathematics), Cam bridge University Press (2009). doi:10.1017/CB09780511702389

work page doi:10.1017/cb09780511702389 2009

[2] [2]

P. A. M. Dirac, K. Fuchs, R. Peierls, and P. D. Preston, Dec lassiﬁed British Report MS-D-5, Part II (1943)

work page 1943

[3] [3]

Underwood, Quantitative Stereology (Addison-Wesle y, 1970)

E. Underwood, Quantitative Stereology (Addison-Wesle y, 1970)

work page 1970

[4] [4]

Serra, Image Analysis and Mathematical Morphology (A cademic Press, London, 1982)

J. Serra, Image Analysis and Mathematical Morphology (A cademic Press, London, 1982)

work page 1982

[5] [5]

Torquato, Random Heterogeneous Materials: microstr ucture and macroscopic properties (Springer-Verlag, New Y ork, 2001)

S. Torquato, Random Heterogeneous Materials: microstr ucture and macroscopic properties (Springer-Verlag, New Y ork, 2001)

work page 2001

[6] [6]

Blanco and R

S. Blanco and R. Fournier, Europhys. Lett. 61, 168 (2003)

work page 2003

[7] [7]

A. Zoia, E. Dumonteil, A. Mazzolo, Phys. Rev. E 85, 011132 (2012); Phys. Rev. E 84, 061130 (2011)

work page 2012

[8] [8]

Mazzolo, A

A. Mazzolo, A. Zoia, and C de Mulatier, Journal of Mathema tical Physics, American Institute of Physics (AIP), 55, 083 308 (2014),and references therein

work page 2014