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arxiv: 2605.16715 · v1 · pith:RZYMQ5ESnew · submitted 2026-05-15 · 🧮 math.CO

Brick Wall Excursions: Combinatorial Interpretation of Random Flight Moments

Sergey Kirgizov , Khaydar Nurligareev , Michael Wallner This is my paper

Pith reviewed 2026-05-20 15:34 UTC · model grok-4.3

classification 🧮 math.CO
keywords random walkslattice pathsDyck pathscombinatorial interpretationmomentsabelian squaresbijectionrandom flights
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The pith

The nth moment of the distance after an m-step random walk equals the number of certain 2n-step lattice paths in dimension m-1 for d=2 and d=4.

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

The paper establishes a combinatorial interpretation for the moments of the expected distance in uniform random walks with unit steps in random directions. It is already known that these moments are integers when the dimension is two or four. The authors construct an explicit count of these moments as the number of 2n-step lattice paths in m-1 dimensions. Their method rests on a bijection that links Dyck paths having a fixed number of peaks to certain words over an m-letter alphabet, and this same map produces the lattice-path objects. A reader cares because the interpretation turns analytic expressions for the moments into direct counting problems that may admit simpler closed forms or recurrences.

Core claim

The authors prove that for dimensions two and four the moments of an m-step random flight are given by the number of 2n-step lattice paths in dimension m-1 that arise from the image of Dyck paths with a prescribed number of peaks under a newly defined bijection to words of a certain type. This bijection additionally supplies closed formulas for the lattice paths equipped with various statistics.

What carries the argument

The bijection between Dyck paths with a prescribed number of peaks and words of a certain type that maps onto 2n-step lattice paths in dimension m-1.

If this is right

  • The construction recovers the known interpretation of the d=2 moments as the number of abelian squares of length 2n over m letters.
  • For d=4 the moments now receive their first combinatorial interpretation as these lattice-path counts.
  • The same bijection yields closed formulas for the number of such lattice paths counted according to various statistics.
  • The interpretation is uniform across the two dimensions and works for arbitrary m and n.

Where Pith is reading between the lines

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

  • If the bijection technique extends, similar counting interpretations might be found for moments in other dimensions.
  • Techniques from the enumeration of lattice paths with peaks could be applied to derive new recurrence relations for the random-walk moments.
  • One could check whether these path counts match known generating functions for the moments beyond the cases already verified.

Load-bearing premise

That the stated bijection between Dyck paths with prescribed peaks and the relevant words or lattice paths exactly preserves the counting of the integer moments of the m-step walk.

What would settle it

For a small fixed m and n, compute the nth moment numerically from the known integral formula for the random flight in d=4 and verify whether it equals the number of 2n-step lattice paths obtained from the bijection construction for that m.

Figures

Figures reproduced from arXiv: 2605.16715 by Khaydar Nurligareev, Michael Wallner, Sergey Kirgizov.

Figure 1
Figure 1. Figure 1: The graph Gm(0): a) for m = 1, b) for m = 2, c) for m = 3. Theorem 1.2. For any two integers m, n ∈ Z>0, the moment Wm+1(0; 2n) is equal to the number of closed paths of length 2n on the graph Gm(0) which start and end at the origin. As we have already mentioned, the particular cases m = 1, 2, 3 of Theorem 1.2 are known: the moments of Wm+1(0; 2n) are already interpreted as paths on certain lattices (see [… view at source ↗
Figure 2
Figure 2. Figure 2: The graph Gm(1): a) for m = 1, b) for m = 2, c) for m = 3. adjacent vertices along the axes x2 and x3 are determined only by the parity of the coordinate sum x1 + x2 + x3. At the same time, one of the adjacent vertices of x ∈ G3(1) depends on the parity of x1 + x2, while another depends on x1 + x2 + x3. More generally, the structure of the adjacent edges of a vertex x ∈ Gm(0) is one of two types: (e1, −e1,… view at source ↗
Figure 3
Figure 3. Figure 3: Bridges of length 4. Peaks are marked red. Given an ith row of the matrix A(0), the sum of its entries is the ith central binomial coefficient (the sequence A000984 in [17]), X i j=0 Aij (0) = X i j=0 i j !2 = [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Dyck paths of length 6. Peaks are marked red. Given an ith row of the matrix A(1), the sum of its entries is equal to the (i + 1)th Catalan number: X i j=0 Aij (1) = X i j=0 N(i + 1, j + 1) = 1 i + 2 2(i + 1) i + 1 ! = Ci+1 . (2.2) The Catalan numbers count Dyck paths and are given by the sequence A000108 in [17]. One more family of lattice paths that will be useful in our investigation is the one of Motzk… view at source ↗
Figure 5
Figure 5. Figure 5: Motzkin path of length 22. The Motzkin paths are enumerated by Motzkin numbers (the sequence A001006 in [17]): Mn = ⌊ n 2X ⌋ k=0 [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Lattice paths from the set P4,1,1. The first three of them are Motzkin paths and form the set Pˆ 4,1,1. Since, for the words of the sets Pn,r,ℓ, the choice of the positions of U’s and D’s is independent, we have the following relations for its cardinality: |P2n,r,ℓ| = [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: The paths on the lattice Z 3 representing the monomials A1 1A −1 2 A1 0A −1 1 A1 0A −1 2 (on the left) and A1 1A −1 2 A1 0A −1 1 A1 2A −1 0 (on the right). Here, m = 2 and n = 3. The actual support of the set of paths corresponding to all possible monomials is the graph (Vm+1, Em+1) whose sets of vertices and edges are defined by Vm+1 = ( (x0, . . . , xm) ∈ Z : Xm k=0 xk = 0 or Xm k=0 xk = 1) and Em+1 = n … view at source ↗
Figure 8
Figure 8. Figure 8: The moments Wm+1(0; 2n) count paths on the graph (Vm+1, Em+1) returning to the origin after 2n steps. The graph is given for a) m = 0, b) m = 1, c) m = 2. Although the graph (Vm+1, Em+1) is defined in the space R m+1, its actual structure is m-dimensional. In the case where m > 0, the orthogonal projection of the graph (Vm+1, Em+1) onto the hyperplane n (x0, . . . , xm) : x0 + . . . + xm = 0o gives us a re… view at source ↗
Figure 9
Figure 9. Figure 9: The moments W3(0; 2n) count paths on the honeycomb lattice returning to the origin after 2n steps. From a counting point of view, the honeycomb lattice (on the left) is equivalent to the lattice made up of bricks (on the right). The deformed lattice is exactly the graph Gm(0), which gives an explanation of Theorem 1.2 based on Proposition 3.1. In the following, we provide a rigorous justification for this … view at source ↗
Figure 10
Figure 10. Figure 10: Bijection. Inserted up steps and down steps are marked blue, appearing peaks are marked red. a) Initial [PITH_FULL_IMAGE:figures/full_fig_p013_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Brick lattice, different settings: a) plane (graph [PITH_FULL_IMAGE:figures/full_fig_p013_11.png] view at source ↗
read the original abstract

We study the expected distance of short uniform random walks in arbitrary dimensions with unit steps in random directions. It is known that for dimensions $d=2$ and $d=4$, all the moments of an $m$-step walk are integer. While for $d=2$, the $n$th moment can be interpreted as the number of abelian squares of length $2n$ over an alphabet with $m$ letters, for $d=4$ no interpretation was known. The goal of this paper is to provide such an interpretation, both for $d=2$ and $d=4$, in terms of $2n$-step lattice paths in dimension $m-1$. Our construction relies on a bijection between Dyck paths with a prescribed number of peaks and words of a certain type. In addition, this bijection allows us to derive closed formulas for the number of lattice paths provided with certain statistics.

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.

Referee Report

1 major / 2 minor

Summary. The paper claims to give combinatorial interpretations of the nth moments of m-step uniform random flights in dimensions d=2 and d=4. For both dimensions the moments are shown to equal the number of 2n-step lattice paths in dimension m-1; the construction proceeds via a bijection that maps Dyck paths with a prescribed number of peaks to words of a certain type, and the same bijection is used to obtain closed formulas for the lattice-path counts under additional statistics.

Significance. If the bijection is shown to preserve the precise statistic that reproduces the known integral expressions for the moments, the work supplies the first explicit counting model for the d=4 case and unifies the d=2 abelian-square interpretation with a lattice-path model. The closed formulas constitute a concrete enumerative contribution that may be useful for further study of random walks and lattice-path generating functions.

major comments (1)
  1. [Section describing the bijection and the d=4 construction] The central claim for d=4 rests on the assertion that the bijection between Dyck paths with a prescribed number of peaks and the relevant words (or lattice paths) exactly preserves the statistic whose generating function equals the moment integral. The manuscript derives closed formulas for the path counts, but does not appear to contain an explicit verification that the image of the peak statistic under the bijection reproduces the integral expression; this verification is load-bearing for the interpretation.
minor comments (2)
  1. Notation for the lattice-path statistics and the precise definition of the target words should be introduced earlier and used consistently throughout the proofs.
  2. A short table comparing the new closed formulas with previously known expressions for small m and n would help the reader assess the formulas.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comment on the d=4 case. We appreciate the positive assessment of the overall contribution and address the major point below.

read point-by-point responses

  1. Referee: [Section describing the bijection and the d=4 construction] The central claim for d=4 rests on the assertion that the bijection between Dyck paths with a prescribed number of peaks and the relevant words (or lattice paths) exactly preserves the statistic whose generating function equals the moment integral. The manuscript derives closed formulas for the path counts, but does not appear to contain an explicit verification that the image of the peak statistic under the bijection reproduces the integral expression; this verification is load-bearing for the interpretation.

    Authors: We agree that an explicit verification of statistic preservation is necessary to make the d=4 interpretation fully rigorous. In the revised manuscript we have added a dedicated paragraph immediately following the definition of the bijection. There we compose the map with the standard generating function for Dyck paths by number of peaks and show, by direct substitution of the peak variable, that the resulting expression coincides term-by-term with the known integral formula for the nth moment in dimension 4. This step confirms that the lattice-path enumeration indeed reproduces the moment and justifies the closed formulas derived from the same bijection. revision: yes

Circularity Check

0 steps flagged

New bijection supplies independent combinatorial interpretation of known integer moments

full rationale

The paper begins from the externally known fact that moments of the m-step random flight are integers when d=2 or d=4. It then constructs an explicit bijection between Dyck paths (with a prescribed peak statistic) and certain words/lattice paths in dimension m-1, using this bijection both to interpret the moments as path counts and to obtain closed formulas for those counts. No equation or count is defined in terms of itself, no parameter is fitted to a subset and then re-labeled as a prediction, and no load-bearing step reduces to a self-citation whose justification is internal to the present work. The derivation is therefore self-contained against the external integral expression for the moments.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard properties of Dyck paths, lattice paths, and the integrality of moments in d=2 and d=4; no free parameters or new invented entities are introduced in the abstract.

axioms (2)
  • domain assumption Moments of the m-step uniform random walk are integers when d=2 or d=4.
    Stated as known in the abstract; used as the starting point for the combinatorial count.
  • ad hoc to paper There exists a bijection between Dyck paths with a prescribed number of peaks and the words or lattice paths that enumerate the moments.
    The construction relies on this bijection; its existence and correctness is the load-bearing step.

pith-pipeline@v0.9.0 · 5696 in / 1367 out tokens · 44579 ms · 2026-05-20T15:34:12.012894+00:00 · methodology

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