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The joint distribution of first return times and of the number of distinct sites visited by a 1D random walk before returning to the origin

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Pith's one-line read The paper derives the exact joint distribution of first return time and number of distinct sites visited for a one-dimensional random walk, yielding closed-form conditional means and variances.

desk verdict Solid closed-form conditional moments for first return and range in 1D, with a fit-based asymptotic section that needs an analytic fix before the subleading terms can be trusted. read the letter →

arxiv 2411.18576 v2 pith:VBTS4K4Z submitted 2024-11-27 cond-mat.stat-mech cond-mat.dis-nn

Mordechai Gruda , Ofer Biham , Eytan Katzav , Reimer Kühn This is my paper · ORCID
classification cond-mat.stat-mechcond-mat.dis-nn MSC 60G5005A1582C4160J10
keywords randomwalkfirstreturntimeDyckpathsCatalannumberjointdistributionconditionalmomentsdistinctsitesvisitedbounded
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

A symmetric random walk on the integers returns to its starting point with probability one, but both the mean first return time and the mean number of distinct sites visited before that return are infinite. This paper shows that conditioning on one of these quantities makes the other finite and explicit: trajectories that visit exactly $s$ distinct sites before returning have a first return time with mean $\frac{2}{3}(s^2+s+1)$ and variance $\frac{4}{45}(s-1)(s+2)(s^2+s-1)$. Conversely, a trajectory whose first return occurs at time $2n$ typically visits about $\sqrt{\pi n}$ distinct sites, with variance growing linearly in $n$. These results are obtained from an exact formula for the joint distribution built from bounded Dyck paths, and they reproduce the known marginal laws as consistency checks. The practical upshot is that first-return search processes can be described by how much territory is covered before the walker comes back.

What carries the argument

The load-bearing object is the combinatorial count $T(n,s)=U(n,s)-U(n,s-1)$, where $U(n,s)$ is the number of bounded Dyck paths of length $2n$ that start and end at zero, never go below zero, and never exceed height $s$; a Dyck path is a walk with steps $\pm1$ that stays nonnegative. Equation (7) expresses $U(n,s)$ as a finite trigonometric sum, and weighting the difference $T(n-1,s-1)$ by the path probability $2^{-(2n-1)}$ produces the joint distribution. The same generating function of $U(n,s)$ supplies the conditional moments through derivatives, and a binomial-coefficient reformulation makes the moments numerically accessible for very large $n$.

What would settle it

A brute-force enumeration of all first-return trajectories for $n$ up to about 20, tallying each by return time $2n$ and maximum height $s$, and comparing the normalized histogram with Eq. (8), would settle the central count; any mismatch for a single $(n,s)$ pair would falsify the joint distribution. The conditional moments could be checked independently by high-statistics simulation at fixed $s=8$.

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Extended reading notes

Core claim

The central discovery is an exact expression for $P(T_{\mathrm{FR}}=2n,S=s)$, the probability that a simple symmetric nearest-neighbor walk on the integers, starting at the origin, first returns at time $2n$ after visiting exactly $s$ distinct sites. The paper writes this probability as $T(n-1,s-1)\,2^{-(2n-1)}$, where $T(n,s)=U(n,s)-U(n,s-1)$ is the number of bounded Dyck paths of length $2n$ whose maximum height is exactly $s$. From this joint distribution it derives the conditional distributions $P(T_{\mathrm{FR}}=2n|S=s)$ and $P(S=s|T_{\mathrm{FR}}=2n)$, and from those the exact conditional mean and variance $\mathbb{E}[T_{\mathrm{FR}}|S=s]=\frac{2}{3}(s^2+s+1)$ and $\mathrm{Var}(T_{\mathrm{FR}}|S=s)=\frac{4}{45}(s-1)(s+2)(s^2+s-1)$, together with the asymptotic laws $\mathbb{E}[S|T_{\mathrm{FR}}=2n]\simeq\sqrt{\pi n}$ and $\mathrm{Var}(S|T_{\mathrm{FR}}=2n)\simeq\pi(\pi/3-1)n$. In this way the divergences of the marginal distributions are controlled, and the coupling between the duration and the spatial extent of a first-return excursion is quantified.

Load-bearing premise

Everything rests on the imported count $U(n,s)$ in Eq. (7), namely that the trigonometric sum equals the number of length-$2n$ walks that stay nonnegative, return to zero, and never exceed height $s$.

Editorial extensions

If this is right

  • Conditioning on territory size removes the divergence: for every finite $s$, the conditional mean and variance of the first return time are exact polynomial functions of $s$.
  • For long excursions the number of distinct sites visited before the first return grows as $\sqrt{\pi n}$, so the explored region extends like the square root of the return time, with a smaller prefactor than the unconditioned mean because the walk must retrace its steps near the end.
  • The conditional distribution $P(T_{\mathrm{FR}}=2n|S=s)$ has an exponential tail whose rate is set by $\cos^2(\pi/(s+1))$, so trajectories that cover more sites are not only slower on average but have return times with a wider spread.
  • The explicit joint distribution reproduces the known marginal laws, including $P(S=s)=1/[s(s+1)]$ and the Catalan-number first-return distribution, which anchors the new formulas to classical one-dimensional random walk results.
  • The binomial-coefficient representation lets the conditional moments be evaluated numerically up to $n=10^6$, making the asymptotic constants directly checkable.

Reading between the lines

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

  • If the same conditioning strategy is applied to biased or resetting walks, whose joint generating functions are already known from previous work, explicit conditional moments of a similar polynomial or square-root form may follow by the same derivative method; this is a natural extension the paper notes but does not carry out.
  • The identification of the number of distinct sites with the maximum height is special to one dimension, so in higher dimensions a genuinely new geometric count is needed and the scaling $\mathbb{E}[S|T]\sim\sqrt{T}$ may not survive.
  • For random-search models in which an agent resets or dies upon returning to the origin, the exact conditional variance quantifies the spread in search outcomes and suggests that the covered territory, rather than elapsed time, is the better control variable.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The paper studies the joint distribution of the first-return time T_FR and the number S of distinct sites visited before first return for a symmetric one-dimensional random walk. Starting from a bounded Dyck path count U(n,s), the authors express the joint distribution P(T_FR=2n,S=s), derive the conditional distributions P(T_FR|S=s) and P(S=s|T_FR=2n), and obtain closed-form conditional moments E[T_FR|S=s]=(2/3)(s^2+s+1) and Var(T_FR|S=s)=(4/45)(s-1)(s+2)(s^2+s-1). For the reverse conditioning, they evaluate exact binomial sums numerically and infer the asymptotics E[S|T_FR=2n] ~ sqrt(pi n), E[S^2|T_FR=2n] ~ (pi^2/3)n - sqrt(pi n), and hence Var(S|T_FR=2n) ~ pi(pi/3-1)n, with additional subleading terms. The paper presents simulation comparisons for the conditional distributions and moments.

Significance. If the exact derivations are correct, the paper advances the recent generating-function result of Klinger et al. by providing an explicit joint distribution and finite conditional moments. The closed-form expressions for E[T_FR|S=s] and Var(T_FR|S=s) are clean, check out for small s, and are supported by simulation, and the leading asymptotics for S|T_FR match known excursion and Theta-distribution results. The main caveat is that the subleading asymptotic terms in Section 7 are obtained by numerical fitting rather than by derivation from the exact sums or from Eq. (42), and Section 8 contains an internal inconsistency in the variance expansion. With those issues corrected, this would be a solid contribution to the random-walk and first-passage literature.

major comments (2)
  1. [Section 7, Eqs. (48)-(57)] The asymptotic results are not derived analytically but are inferred by evaluating the exact sums (47) and (52) at finite n and fitting the residuals Delta_1 and Delta_2 to assumed decay forms (Eqs. (50) and (56)). No derivation is given from the exact combinatorial sums or from the Theta distribution in Eq. (42). The leading constants are plausible and agree with known results, but the subleading coefficients in Eqs. (51) and (57), which are advertised as new, are fit-dependent and not established. These terms should either be derived analytically or presented explicitly as numerical conjectures with a clear statement of their status.
  2. [Section 8, variance expansion] The displayed asymptotic Var(S|T_FR=t) ~ (pi^2/6 - pi/2)t - sqrt(pi/2) sqrt(t) + O(1) is inconsistent with combining Eqs. (51) and (57). From those equations, E[S|t]^2 = (pi/2)t - sqrt(pi/2) sqrt(t) + 1/4 + ..., so the sqrt(t) terms cancel in Var(S|T_FR=t) = E[S^2|t] - E[S|t]^2, leaving Var(S|T_FR=t) ~ (pi^2/6 - pi/2)t - (pi^4/450 + 1/4) + ... . The stated sqrt(t) term in Section 8 therefore cannot be correct, and this internal inconsistency indicates that the subleading expansion is not fully controlled.
minor comments (4)
  1. [Section 6, Eq. (39)] Eq. (39) appears to have an index-shift error: substituting U(n-1,s-1)-U(n-1,s-2) into Eq. (14) yields a prefactor 2^{2n-1} with arguments (n-1,s-1), (n,s-1), (n-1,s-2), (n,s-2), not the displayed F(n,s), F(n+1,s), F(n,s-1), F(n+1,s-1) with prefactor 2^{2n+1}. The current form is not equal to Eq. (37) and can take negative values, for example for n=2, s=1.
  2. [Section 8, biased random walk discussion] The statements that for p<1/2 the walk is recurrent with P_R=1 and finite mean first-return time, and that for p>1/2 the return probability is (1-p)/p, are only correct for the process conditioned on the first step being to the right. Unconditioned, a biased one-dimensional random walk with p != 1/2 is transient. Please clarify that the conditioning on x_1=1 is maintained throughout this paragraph.
  3. [Section 7, Eq. (54)] Eq. (54) cannot be valid for all moments as stated: for r=1 the formula gives zero, whereas E[S|T_FR=2n] ~ sqrt(pi n). The displayed leading-order formula should be restricted to r>=2, or supplemented by the separate r=1 result.
  4. [Appendix B] Appendix B uses the word 'nodes' in one place where the rest of the paper uses 'sites'; please unify the terminology.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: exact conditional moments follow from the externally cited combinatorial count and the generating function; Section 7 asymptotics are numerical extrapolations, not circular reductions.

full rationale

The central derivation is self-contained. The joint distribution (Eq. 8) is built from the bounded-Dyck-path count U(n,s) (Eq. 7), which is imported from de Bruijn et al. [45] and Hein & Huang [31] - not from the present authors' prior work - and is cross-checked against the binomial representation (A.4). The conditional moments E[T|S=s] and Var(T|S=s) (Eqs. 32 and 35) are obtained by differentiating the generating function Rs(x) (Eqs. 29-31) at x=0; no fitted parameter is fed back into the calculation. The asymptotic statements of Section 7 are obtained by numerically evaluating the exact finite-n expressions (47) and (52) and fitting the residuals Delta_1 and Delta_2 (Eqs. 50 and 56) to assumed decay laws, so those subleading constants are heuristic rather than proven; nevertheless this is an extrapolation of an exact expression, not a reduction of a predicted quantity to its own input. The Section 8 claim Var(S|T=t) ~ (pi^2/6 - pi/2)t - sqrt(pi/2) sqrt(t) + O(1) appears inconsistent with combining Eqs. (51) and (57), where the sqrt(t) terms cancel; this is an internal consistency error in the subleading asymptotics, not a circularity. Self-citations (e.g., Tishby et al. on random regular graphs) concern different models and are not load-bearing. Score 0.

Assumptions & free parameters 2 free parameters · 3 assumptions · 0 invented entities

The paper introduces no new entities. Its free parameters are the fitted coefficients of the asymptotic scaling forms in Sec. 7, which are subsequently identified with known constants. The main load-bearing axiom is the imported formula for bounded Dyck paths, Eq. (7). The domain assumption s = max height is valid for the 1D first-return setting. The ad hoc scaling forms weaken the asymptotic part of the derivation.

free parameters (2)
  • Asymptotic coefficient a in E[S|TFR=2n] ~ a sqrt(n) = sqrt(pi)
    Obtained by fitting high-precision evaluations of Eq. (47) to an assumed scaling form; consistent with known average height of Dyck paths.
  • Asymptotic coefficient b in E[S^2|TFR=2n] ~ b n = pi^2/3
    Fitted from Eq. (52) and matched to the known Flajolet-Sedgewick leading moment for r=2.
assumptions (3)
  • standard math The exact count of Dyck paths bounded by height s is U(n,s) = (2^{2n+1}/(s+2)) sum_{m=1}^{s+1} sin^2(m pi/(s+2)) cos^{2n}(m pi/(s+2)), as stated in Eq. (7).
    Imported from de Bruijn-Knuth-Rice and Hein-Huang; not proven in this paper. Used to derive the joint distribution.
  • domain assumption For a first-return trajectory starting with a step to the right, the number of distinct sites visited equals the maximum height of the path, so s = max height.
    Used in Sec. 3 to map S to Dyck path height; valid because the path never crosses to the negative side.
  • ad hoc to paper Asymptotic scaling forms assumed for E[S|TFR=2n] ~ a sqrt(n) + b + c/sqrt(n) and E[S^2|TFR=2n] ~ a2 n + b2 sqrt(n) + c2.
    Used in Sec. 7 to fit the high-precision numerical evaluation of Eqs. (47) and (52); the forms are plausible but not proven analytically.

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Pith. "Pith review of The joint distribution of first return times and of the number of distinct sites visited by a 1D random walk before returning to the origin." pith.science (2026). https://pith.science/paper/VBTS4K4Z

@misc{pith2026241118576,
  author       = {Pith},
  title        = {Pith review of: The joint distribution of first return times and of the number of distinct sites visited by a 1D random walk before returning to the origin},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VBTS4K4Z}},
  note         = {Machine review of arXiv:2411.18576}
}
abstract

We present analytical results for the joint probability distribution $P(T_{FR}=t,S=s)$ of first return (FR) times t and of the number of distinct sites s visited by a random walk (RW) on a one dimensional lattice before returning to the origin. The RW on a one dimensional lattice is recurrent, namely the probability to return to the origin is $P_{R}=1$. However the mean $\langle T_{FR}\rangle$ of the distribution $P(T_{FR}=t)$ of first return times diverges. Similarly, the mean $\langle S\rangle$ of the distribution $P(S=s)$ of the number of distinct sites visited before returning to the origin also diverges. The joint distribution $P(T_{FR}=t,S=s)$ provides a formulation that controls these divergences and accounts for the interplay between the kinetic and geometric properties of first return trajectories. We calculate the conditional distributions $P(T_{FR}=t|S=s)$ and $P(S=s|T_{FR}=t)$. We find that the conditional expectation value of first return times of trajectories that visit s distinct sites is ${\mathbb E}[T_{FR}|S=s]=\frac{2}{3}(s^2+s+1)$, and the variance is $Var(T_{FR}|S=s)=\frac{4}{45}(s-1)(s+2)(s^2+s-1)$. We also find that in the asymptotic limit, the conditional expectation value of the number of distinct sites visited by an RW that first returns to the origin at time $t=2n$ is ${\mathbb E}[S|T_{FR}=2n] \simeq \sqrt{\pi n}$, and the variance is $Var(S|T_{FR}=2n) \simeq \pi\left(\frac{\pi}{3}-1\right)n$. These results go beyond the important recent results of Klinger et al. [{\it Phys. Rev. E} {\bf 105}, 034116 (2022)], who derived a closed form expression for the generating function of the joint distribution, but did not go further to extract an explicit expression for the joint distribution itself. The joint distribution provides useful insight on the efficiency of random search processes, in which the aim is to cover as many sites as possible in a given number of steps.

Figures

Figures reproduced from arXiv: 2411.18576 by the authors.

Figure 1
Figure 1. Illustration of a first return trajectory of an RW on a one-dimensional lattice. It shows the location xt of the RW as a function of time, starting from the origin at t = 0 until it returns to the origin for the first time. The number s of distinct sites visited by the RW before returning to the origin (not including the origin itself) is given by the maximum value of xt. of the number of distinct sites visited by t… view at source ↗
Figure 2
Figure 2. Analytical results (solid line) for the conditional probability distribution P(TFR = t|S = s) of first return times t = 2n of RWs that have visited s = 8 distinct sites before returning to the origin for the first time. The analytical results, obtained from Eq. (24), are in very good agreement with the results obtained from computer simulations (circles). E[T r FR|S = s] = X∞ n=1 (2n) rP(TFR = 2n|S = s). (26) To thi… view at source ↗
Figure 3
Figure 3. Analytical results (solid line) for the expectation value E[TFR|S = s] of the first return time of RWs that have visited s distinct sites before returning to the origin for the first time, as a function of s. The analytical results, obtained from Eq. (32), are in very good agreement with the results obtained from computer simulations (circles). Inserting Rs(x) from Eqs. (30) and (10) into the right hand side of Eq. … view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: Analytical results (solid line) for the variance Var(TFR|S = s) of the conditional distribution of first return times of RWs that have visited s distinct sites before returning to the origin for the first time, as a function of s. The analytical results, obtained from …
Figure 5
Figure 5. Figure 5: Analytical results (solid line) for the conditional probability distribution P(S = s|TFR = t) of the number of distinct sites visited by RWs whose first return time to the origin is t = 80. The analytical results, obtained from Eq. (37), are in very good agreement with…
Figure 6
Figure 6. Figure 6: Analytical results (solid line) for the conditional expectation value E[S|TFR = t] of the number of distinct sites visited by an RW whose first return time to the origin is t. The analytical results, obtained from Eq. (47), are in very good agreement with the results o…
Figure 7
Figure 7. Figure 7: Analytical results (+ symbols) for the difference ∆1, given by Eq. (49), as a function of the time t. These results are obtained by direct numerical evaluation of E[S|TFR = t], using Eq. (47). These results are well fitted by Eq. (50), represented by the dashed line. W…
Figure 8
Figure 8. Figure 8: Analytical results (+ symbols) for the difference ∆2, given by Eq. (55), as a function of the time t. The analytical results are obtained by direct numerical evaluation of E[S 2 |TFR = t], using Eq. (52). These results are well fitted by Eq. (56), represented by the da…
Figure 9
Figure 9. Figure 9: Analytical results (solid line) for the variance Var(S|TFR = t) of the distribution of the number of distinct sites visited by an RW whose first return time to the origin is t. The analytical results, obtained from Eq. (58), are in very good agreement with the results …

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