Exact closed-form recurrence probabilities for biased random walks at any step number

Debendro Mookerjee; Sarah Kostinski

REVIEW 2 major objections 5 minor 18 references

Exact closed-form recurrence probabilities for biased random walks at any step number

T0 review · 2 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read This paper derives exact, closed-form survival probabilities for a biased one-dimensional random walk valid at every step number, and uses them to show that above a finite-size critical bias the probability of last return decays…

desk verdict Exact finite-N survival formulas are the real contribution; the critical-bias monotonicity claim is not proven. read the letter →

arxiv 2412.17994 v1 pith:6ARJDVZJ submitted 2024-12-23 cond-mat.stat-mech physics.data-an

Debendro Mookerjee , Sarah Kostinski This is my paper · ORCID

classification cond-mat.stat-mechphysics.data-an MSC 60G5082B41 PACS 05.40.Fb

keywords biasedrandomwalksurvivalprobabilitylastreturnfirstclosed-formexpressionhypergeometricfunctionarcsinelawcriticalbias

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 derives exact formulas for the probability that a biased one-dimensional random walker never returns to its starting point within $N$ steps, for any step number and any bias $B=p-q$. Known asymptotic results cover only the long-walk limit, where the survival probability becomes $|B|$; here the finite-$N$ regime is included. From these formulas the authors obtain the probability that the last return occurs at a given step, and they show that beyond a bias threshold $|B_c|=\sqrt{(n-1)/(3n-1)}$ the last-return probability decreases monotonically throughout a $2n$-step walk, with the threshold approaching $1/\sqrt{3}$ for long walks. The closed forms matter for intermediate-length walks, such as those taken by molecular motors along microtubules, where large-$N$ approximations would miss the monotone regime.

What carries the argument

The argument is carried by the closed-form survival probability $S(N,B)$ built from ballot-theorem path counting. For a sequence with $N_+$ right steps and $N_-$ left steps, the fraction of paths that stay strictly positive is $(N_+-N_-)/N$; summing $p^{N_+}q^{N_-}$ times this factor over all $N_-$ gives the probability of surviving on the right, and the same sum with $p$ and $q$ swapped gives the probability of surviving on the left. Adding the two and rewriting in terms of $B=p-q$ produces Eq. 3, which the authors evaluate as a terminating ${}_2F_1$ hypergeometric series to obtain Eq. 4a and Eq. 4b. All later quantities are derivatives of this closed form: the first-return distribution $F(N)$ from the identity $F(i+1)=S(i)-S(i+1)$, the probability of last return $PLR(x,B)$ as the product of the probability of being at the origin at fraction $x$ and the subsequent survival probability, and the critical bias from the endpoint inequality $PLR(1) < PLR(1-1/n)$.

What would settle it

Enumerate all $2^N$ step sequences for small $N$ (for example $N=20$) at several biases and compare the exact survival probability with Eq. 4; then evaluate $PLR(x,B)$ from Eq. 8 at every $x=k/n$ for a $2n=200$ step walk with $B$ just above $\sqrt{199/599}$, checking that the sequence is nonincreasing. Any mismatch would falsify the closed forms, and any rise at an intermediate $x$ would falsify the monotone claim above $B_c$.

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Extended reading notes

Core claim

The central claim is that Eq. 4a and Eq. 4b give the exact survival probability $S(N,B)$ of a discrete walker with bias $B=p-q$, valid for every $N\geq1$ and every $-1<B<1$. The formulas are written in terms of the hypergeometric function ${}_2F_1$, with separate cases for odd and even $N$. The authors derive them by counting surviving paths with the ballot theorem, summing probabilities over all possible numbers of backward steps, and converting the finite sum into a closed form. Taking differences of survival probabilities yields the first-return distribution, and multiplying the probability of being at the origin at step $2n_L$ by the survival probability for the remaining steps gives the probability of last return (Eq. 8). At zero bias the last-return distribution reproduces the arcsine law, and as $N$ grows the survival probability approaches $|B|$. The paper's new finite-size result identifies a critical bias $|B_c|=\sqrt{(n-1)/(3n-1)}$ above which the last-return probability decreases monotonically throughout a $2n$-step walk; this threshold saturates at $1/\sqrt{3}$ for infinitely long walks.

Load-bearing premise

The conclusion that the last-return probability is monotone decreasing above $B_c$ rests on the assumption that comparing only the final two points of the walk, $PLR(1) < PLR(1-1/n)$, is enough to guarantee monotonicity at every earlier step; the paper does not prove or numerically check monotonicity for intermediate values of $x$.

Editorial extensions

If this is right

For any finite $N$ and bias $B$, the survival probability $S(N,B)$ is exactly computable, so intermediate-length walks no longer require large-$N$ approximations.
The first-return distribution follows by differencing, $F(i+1)=S(i)-S(i+1)$, vanishes automatically on odd steps, and exhibits the $N^{-3/2}$ power law only in the unbiased case.
The cumulative last-return probability saturates at $1-|B|$, and the arcsine law appears only at zero bias; any nonzero bias breaks the symmetric, bimodal last-return shape.
For a $2n$-step walk, biases above $\sqrt{(n-1)/(3n-1)}$ put the last-return probability in the monotone regime, with the threshold approaching $1/\sqrt{3}\approx0.577$ for long walks.
Molecular motors with effective bias around 0.6--0.8 over about 100 steps sit above this threshold, so their recurrence and last-return statistics follow the monotone regime described here.

Reading between the lines

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

A testable extension: evaluate Eq. 8 at all intermediate $x$ for $B$ just above $B_c$ to confirm the monotone claim holds globally and not only at the final two steps.
A consequence the paper leaves implicit: because Eq. 4 is exact, finite-size corrections to the arcsine law can be written down, for example a $B$-dependent deformation of the cumulative last-return distribution interpolating between $\arcsin\sqrt{x}$ and the step function as $|B|\to1$.
A neighbouring problem: the same ballot-counting strategy could yield finite-$N$ closed forms for walks with reflecting or absorbing boundaries.

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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

Quick take: the exact finite-N survival formulas in Eq. 4 are solid and genuinely useful; the critical-bias monotonicity claim is the weak spot and should not be stated as proven.

The paper's real contribution is the closed-form hypergeometric expressions for survival probability of a biased 1D walk for arbitrary N. That is not in the cited literature as far as I can tell, and the derivation via ballot-theorem sums is straightforward and credible. The formulas reproduce the unbiased limit, the large-N S=|B|, and the Monte Carlo data in Figs. 1 and 4. The PLR formula in Eq. 8 is also exact and gives the arcsine law at B=0. The paper is clearly written, and the self-citation [9] is just a standard ballot-theorem reference, so no circularity concern.

The soft spot is the critical-bias result. B_c is derived by requiring PLR(1,B) < PLR(1-1/n,B), i.e. only the final gap. The text then asserts that above B_c the PLR decays monotonically throughout the walk. That does not follow from the endpoint inequality. For B=0 the PLR is U-shaped with two peaks; fixing the right endpoint doesn't rule out an interior local maximum. No proof or numerical scan for intermediate x is given. So the stated critical bias is really the threshold for disappearance of the late peak, not for global monotonicity. This is a genuine gap in a headline claim, though it may well be true; the fix is either a proof (e.g. showing PLR(x+1/n) < PLR(x) for all x) or softening the wording.

Minor issue: the derivation of Eq. 4 is relegated to a supplementary that isn't in this version. That's fine for a preprint but should be available for refereeing.

Who is this for? People working on finite-time first-passage or return statistics, and the molecular motor discussion is a plausible application. The exact intermediate-N results are worth having.

Recommendation: send to peer review. The core formulas are valuable and likely correct, but the monotonicity claim needs to be either proven or scaled back before publication.

Referee Report

2 major / 5 minor

Summary. This paper treats the discrete-time, one-dimensional biased random walk and derives closed-form expressions for the survival probability (the probability of not returning to the origin) for any finite number of steps N. Equation 4 gives the result in terms of a Gauß hypergeometric function for odd and even N; the authors verify it against Monte Carlo simulations and recover the unbiased binomial result and Pólya's large-N limit S=|B|. From the survival probability they obtain the first-return distribution via a difference, and the probability of last return (PLR) as a product of the probability of being at the origin and the subsequent survival probability. They then define a critical bias |B_c| = sqrt((n-1)/(3n-1)) above which the PLR is claimed to decay monotonically, and they discuss applications to molecular motors.

Significance. The work provides exact finite-N expressions for a classical model, which fills a genuine gap between known short- and long-time treatments. The derivations appear to use the ballot theorem and a hypergeometric summation identity, and the numerical validation in Figs. 1 and 4 supports the survival-probability formula. No parameters are fitted. The advertised qualitative findings—the monotonic decay of the PLR beyond a critical bias and the saturation of the critical bias at 1/sqrt(3)—are of clear physical interest, particularly for molecular-motor processivity. The main deficit is that the monotonicity statement is not proven or numerically established for intermediate x; this is the load-bearing part of the abstract's central claim.

major comments (2)

[Probability of last return and critical bias (Eqs. 9–10, Fig. 6)] The critical-bias condition is derived only from the endpoint comparison PLR(1,B) < PLR(1-1/n,B). Monotonic decrease of the PLR in x for all x requires positivity of every discrete increment PLR(k/n,B) - PLR((k+1)/n,B) for k=0,...,n-1. The manuscript does not prove this nor provide a numerical scan of all gaps. Consequently, the sentence 'Above this critical bias, a walk of total step number 2n will have a monotonically decreasing PLR' is not justified by the derivation shown. I request either a proof of monotonicity (e.g., by showing the sign of the x-derivative of Eq. 8) or an explicit numerical verification for representative n and B just above B_c, and a revised statement if monotonicity fails.
[Probability of last return and critical bias (Eq. 10) and molecular-motor discussion] The universal statement that B > 1/sqrt(3) guarantees a monotonically decreasing PLR for every walk length inherits the gap described above. The saturation argument shows only that the endpoint inequality holds for all n when B > 1/sqrt(3); it says nothing about interior values of x. Since the molecular-motor application (dynein biases of 0.6–0.8 versus B_c ≈ 0.573 for ~100 steps) relies directly on this monotonicity, the claim needs to be either proved or softened to 'the endpoint peak at x=1 disappears.'

minor comments (5)

[Eq. 4a] The statement 'These expressions hold for any step number (N ≥ 1)' should be checked for N=1; the expression as typeset appears to give S(1,0)=0.5 rather than 1. If the formula is intended only for N≥3, this should be stated explicitly.
[Fig. 5] In panel (a), it would be helpful to mark the critical-bias curve and to state clearly that the curves are computed from Eq. 8; no simulation data are shown for the PLR, so the agreement is not directly visible for this quantity.
[Eq. 3 and text after it] The statement that the survival probabilities for an even step number and its consecutive odd step number are the same is true for N≥2; N=1 is a special case (S(1)=1, S(2)=(1+B^2)/2) and should be excluded.
[References] Reference [11] is dated 1942; the commonly cited edition of Abramowitz and Stegun is from 1964 or later, so please correct the date.
[Abstract and intro] The name 'Pólya' is typeset in one place as 'Poly´a'; please fix the diacritics throughout.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation: the closed-form survival formula is derived independently from the ballot theorem and a hypergeometric identity; the only self-citation is a standard ballot-theorem reference, and no fitted quantity is relabeled as a prediction.

full rationale

The central result, Eq. 4, is obtained by applying the ballot theorem to enumerate paths whose running sum stays positive or negative and then summing the resulting binomial terms into a Gauss hypergeometric function. The ballot theorem is cited to Refs. [8,9]; Ref. [9] is by one of the present authors, but it supplies the standard combinatorial fact (N_+ - N_-)/N, not the survival probability or the critical-bias result under test, so the self-citation is not load-bearing. The probability-of-last-return formula, Eq. 8, is constructed as the product of the probability of being at the origin at step 2n_L and the independently derived survival probability for the remaining steps; this is a standard last-return decomposition, not a tautology. The large-N limits S=|B| and the arcsine law are recovered from the closed forms, not assumed as inputs. No parameter is fitted to data and then renamed a prediction; simulations serve only as external checks. The one non-circular gap is the critical-bias inference: Eq. (10) is obtained solely from the endpoint inequality PLR(1,B) < PLR(1-1/n,B) in Eq. (9), while the paper claims monotone decrease of the PLR for all intermediate x above B_c. Monotone decrease requires every adjacent gap PLR(k/n)-PLR((k+1)/n) to be positive, not merely the final gap, and that is not proved or numerically scanned. This is an unsupported leap and therefore a correctness risk, not a circularity, because the endpoint condition is not equivalent by construction to monotonicity. The molecular-motor discussion relies on this unproven monotonicity, but that does not make the derivation circular. Overall, the paper is self-contained against external combinatorial identities and simulations, so the circularity burden is minimal.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters are fitted anywhere in the paper; p, q, N are given inputs. The derivation relies on four items: ballot theorem, parity of returns, Markov factorization for the last-return probability, and a hypergeometric summation identity. No new physical entities are introduced.

assumptions (4)

standard math Ballot theorem: among sequences with N_+ > N_-, the fraction that remain strictly positive is (N_+ - N_-)/N.
Used in Eq. 1 to count paths that never return to zero; this is a classical combinatorial result, independent of p and q.
domain assumption The survival probability S(2n) equals S(2n+1) because returns to the origin can only occur at even step numbers.
Invoked after Eq. 4 to justify computing only Seven; it follows from parity of the lattice walk.
domain assumption The probability of last return at 2nL equals the probability of being at the origin at 2nL times the survival probability for the next 2n-2nL steps.
Used to build Eq. 8; relies on the Markov property of the walk and independence of future steps from the path history.
standard math The sum in Eq. 3 can be evaluated in closed form as a Gauss hypergeometric function.
Passage from Eq. 3 to Eq. 4; details are deferred to the supplementary material.

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Pith. "Pith review of Exact closed-form recurrence probabilities for biased random walks at any step number." pith.science (2026). https://pith.science/paper/6ARJDVZJ

@misc{pith2026241217994,
  author       = {Pith},
  title        = {Pith review of: Exact closed-form recurrence probabilities for biased random walks at any step number},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6ARJDVZJ}},
  note         = {Machine review of arXiv:2412.17994}
}

abstract

We report on a closed-form expression for the survival probability of a discrete 1D biased random walk to not return to its origin after N steps. Our expression is exact for any N, including the elusive intermediate range, thereby allowing one to study its convergence to the large N limit. In that limit we recover Polya's recurrence probability, i.e. the survival probability equals the magnitude of the bias. We then obtain a closed-form expression for the probability of last return. In contrast to the bimodal behavior for the unbiased case, we show that the probability of last return decays monotonically throughout the walk beyond a critical bias. We obtain a simple expression for the critical bias as a function of the walk length, and show that it saturates at $1/\sqrt{3}$ for infinitely long walks. This property is missed when using expressions developed for the large N limit. Finally, we discuss application to molecular motors' biased random walks along microtubules, which are of intermediate step number.

Figures

Figures reproduced from arXiv: 2412.17994 by the authors.

**Figure 2.** FIG. 2. Convergence properties of [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 4.** compares the analytic results for even N to simulations (see Supplementary Material [10] for F(N) formulae). In the case of a symmetric walk (B = 0), one recovers the power law behavior ∼N −3/2 , as shown in blue in [PITH_FULL_IMAGE:figures/full_fig_p003_4.png] view at source ↗

Figures from the paper (2 more)

**Figure 5.** Figure 5: FIG. 5. Panel (a): Plot of the probability of last return (PLR) to the origin at step 2 [PITH_FULL_IMAGE:figures/full_fig_p004_5.png]

**Figure 6.** Figure 6: FIG. 6. The critical bias [PITH_FULL_IMAGE:figures/full_fig_p004_6.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

18 extracted references · 17 canonical work pages

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The fraction of such paths whose running sum remains above 0 is given by the ballot theorem [8, 9]:(N+ − N−)/N = 1 − 2N−/N. We sum the probabilities of all such paths to obtain R(N): R(N) = ⌊ N−1 2 ⌋ ∑ N−=0 pN−N− qN− 1 − 2N− N N N− (1) where we have chosen the limits such thatN+ > N−, or equiv- alently, 0 ≤ N− ≤ ⌊N−1 2 ⌋. This expression can be simplified...

work page Pith review arXiv 2024
[2]

This is in agreement with the fact that return can occur only at even step numbers

is simply zero at odd step numbers. This is in agreement with the fact that return can occur only at even step numbers. Fig. 4 compares the analytic results for even N to simulations (see Supplementary Material [10] for F(N) formulae). In the case of a symmetric walk (B = 0), one recovers the power law behavior ∼ N−3/2, as shown in blue in Fig. 4. We now ...

work page
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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [9]

Feller, W., An Introduction to Probability Theory and Its Appli- cations, 3rd ed. , V ol. 1 (John Wiley & Sons, New York) 1967. For the cited quote, see Section XI.4 on partial fraction expan- sions

work page 1967

[2] [1]

The fraction of such paths whose running sum remains above 0 is given by the ballot theorem [8, 9]:(N+ − N−)/N = 1 − 2N−/N. We sum the probabilities of all such paths to obtain R(N): R(N) = ⌊ N−1 2 ⌋ ∑ N−=0 pN−N− qN− 1 − 2N− N N N− (1) where we have chosen the limits such thatN+ > N−, or equiv- alently, 0 ≤ N− ≤ ⌊N−1 2 ⌋. This expression can be simplified...

work page Pith review arXiv 2024

[3] [2]

This is in agreement with the fact that return can occur only at even step numbers

is simply zero at odd step numbers. This is in agreement with the fact that return can occur only at even step numbers. Fig. 4 compares the analytic results for even N to simulations (see Supplementary Material [10] for F(N) formulae). In the case of a symmetric walk (B = 0), one recovers the power law behavior ∼ N−3/2, as shown in blue in Fig. 4. We now ...

work page

[4] [3]

Ann., 84 (1921)

P ´olya G., Math. Ann., 84 (1921)

work page 1921

[5] [4]

and Snell, J.L

Doyle, P.G. and Snell, J.L. Random Walks and Electric Net- works, 1st ed. (Mathematical Association of America) 1984

work page 1984

[6] [5]

and Sokolov, I.M., First Steps in Random Walks: From Tools to Applications (Oxford University Press Inc., New York) 2011

Klafter, J. and Sokolov, I.M., First Steps in Random Walks: From Tools to Applications (Oxford University Press Inc., New York) 2011

work page 2011

[7] [6]

Redner, S., A Guide to First-Passage Processes (Cambridge University Press, New York) 2001

work page 2001

[8] [7]

and Majumdar, S., Phys

Chupeau, M., B ´enichou, O. and Majumdar, S., Phys. Rev. E, 91 (2015) 032106

work page 2015