Survival probabilities in biased random walks: To restart or not to restart? that is the question

Shahar Hod

REVIEW 1 major objections 4 minor 23 references

Survival probabilities in biased random walks: To restart or not to restart? that is the question

T0 review · 1 major / 4 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read A biased walker with random restarting outlives an ordinary one only when it starts beyond a $q$-dependent critical gap $x_0^{\mathrm{crit}}(q)$.

desk verdict The threshold values in Table II and Eq. (19) are wrong because the dominant characteristic root is canceled by the initial conditions; the qualitative threshold idea survives, but the paper's quantitative claims do not. read the letter →

arxiv 2502.06667 v1 pith:OKLXWLUQ submitted 2025-02-10 cond-mat.stat-mech math-phmath.MP

Shahar Hod This is my paper · ORCID

classification cond-mat.stat-mechmath-phmath.MP MSC 60J10 PACS 05.40.Fb

keywords biasedrandomwalkSisyphusstochasticresettingsurvivalprobabilityabsorbingboundarycriticalinitialgapexponentialdecayrenewalequation

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

The paper asks whether a random walker that at each step either moves toward an absorbing trap or is reset to its starting position has a better chance of long-time survival than an ordinary walker with the same bias toward the trap. The answer it establishes is a threshold: the Sisyphus walker's survival probability decays exponentially at rate $\beta$, the largest positive root of $\beta^{x_0+1}-\beta^{x_0}+q^{x_0}(1-q)=0$, and for large $x_0$ this rate is approximately $1-q^{x_0}(1-q)$. Comparing this rate with the standard walker's $2\sqrt{q(1-q)}$ yields a critical initial gap $x_0^{\mathrm{crit}}(q)$ above which the restarting walker survives longer and below which it survives shorter. The interest is that resetting is neither automatically good nor automatically bad: its value depends on where the walker starts.

What carries the argument

The central object is the linear recurrence $S(t;x_0,q)=S(t-1;x_0,q)-q^{x_0}(1-q)\,S(t-x_0-1;x_0,q)$, obtained by counting the walkers who are absorbed only after $x_0$ consecutive steps toward the trap and those who reset from $x_0$ before it is too late. Substituting the exponential ansatz $S(t)=\alpha\beta^t$ turns this recurrence into the characteristic polynomial $\beta^{x_0+1}-\beta^{x_0}+q^{x_0}(1-q)=0$; the largest positive root of this polynomial is the asymptotic decay factor per time step. A first-order expansion of that root in the large-$x_0$ regime gives the closed-form rate $1-q^{x_0}(1-q)$ that is compared with the standard walker's rate $2\sqrt{q(1-q)}$.

What would settle it

Evaluate the largest positive root $\beta$ of $\beta^{5}-\beta^{4}+(0.7)^4(0.3)=0$: Eq. (17) predicts $\beta\approx0.92797>0.9165=2\sqrt{0.7\cdot0.3}$, while the exact root is $\beta\approx0.8798<0.9165$, so iterating the recurrence (9) at $x_0=4$, $q=0.7$ would show the standard walker, not the Sisyphus walker, has the larger late-time survival probability.

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Extended reading notes

Core claim

For biased Sisyphus random walkers — particles on non-negative integers that move one step toward the absorbing origin with probability $q>1/2$ and jump back to their initial position $x_0$ with probability $1-q$ — the paper derives the survival probability recurrence $S(t)=S(t-1)-q^{x_0}(1-q)\,S(t-x_0-1)$ for $t>x_0$. It shows that the late-time solution is $S(t)\sim \alpha\, \beta^t$, with $\beta$ determined by the characteristic polynomial $\beta^{x_0+1}-\beta^{x_0}+q^{x_0}(1-q)=0$, and evaluates the prefactor $\alpha$ in the large-$x_0$ regime. Since the standard biased walker has asymptotic ratio $S(t+1)/S(t)\to 2\sqrt{q(1-q)}$, the Sisyphus walker's ratio $1-q^{x_0}(1-q)$ (for $x_0,t\gg1$) crosses this baseline at $x_0^{\mathrm{crit}}(q)=\ln[(1-2\sqrt{q(1-q)})/(1-q)]/\ln q$. The paper therefore claims a critical gap that separates restarting from not restarting, with the threshold diverging as $q\to1/2$ and a uniform dominance of the Sisyphus walker for $q\gtrsim0.78$.

Load-bearing premise

The closed-form critical-gap formula (17) rests on the approximation $\beta\approx1-q^{x_0}(1-q)$, which is accurate only when $x_0 q^{x_0}(1-q)\ll1$; the paper uses it in regimes such as $q=0.7$, $x_0=4$ where that condition fails, so the exact polynomial criterion yields a different threshold.

Editorial extensions

If this is right

For a fixed bias $q$, the late-time ranking between the restarting and non-restarting walkers is decided solely by whether the starting gap $x_0$ lies above or below $x_0^{\mathrm{crit}}(q)$.
The critical gap grows without bound as the bias weakens, diverging as $-2\ln(q-1/2)/\ln2-2$ when $q\to1/2^+$, so the restart strategy only helps from very large initial distances near the unbiased limit.
For every $q>0.78$, the resetting walker has the larger asymptotic survival probability for every possible integer starting position $x_0\ge1$.
The survival probability of the Sisyphus walker is exponentially decaying with a $q$- and $x_0$-dependent rate, so the restart mechanism changes the decay rate, not merely the prefactor of the survival tail.

Reading between the lines

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

Editorial: An exact threshold can be defined by solving $\beta(x_0,q)=2\sqrt{q(1-q)}$ with $\beta$ the largest root of the characteristic polynomial; using that exact criterion instead of the first-order expansion shifts some table entries by one step (for example, $x_0=4$, $q=0.7$) and removes the smallness assumption behind Eq. (17).
Editorial: The same recurrence can be solved by generating functions to produce the full time-dependent survival probability $S(t;x_0,q)$ for all $t$, not just its asymptotic exponential tail, and the resulting expression could be compared directly with Monte Carlo iteration of the jump rule.
Editorial: The threshold phenomenon is a specific instance of a general restart trade-off: resetting helps when the walker already starts in a region that is safe enough relative to the trap, and hurts when it must be free to reach large distances; introducing a cost per reset would move the critical gap upward.

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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

The short version: the paper's central quantitative claim is incorrect. The critical-gap values in Table II and the q>0.78 claim in Eq. (19) are false for small x0.

What the paper does well: the discrete recurrence (9) is correctly derived, the exponential-decay ansatz is standard, and the numerical check in Table I (x0=10, q=3/4) is solid. The paper is clearly written and the large-x0 analysis is fine. The model itself is not new, and the linear recurrence solution is standard, but the explicit comparison with the ordinary biased walker is a legitimate extension.

The problem is that the characteristic polynomial (12) always has β=q as a root. For small x0, this is the largest root, but it is canceled by the initial conditions. The actual decay rate is the largest root of the reduced polynomial Q(λ)=λ^{x0}−(1−q)(λ^{x0−1}+qλ^{x0−2}+...+q^{x0−1}). For x0=1, q=0.9, the paper's approximation gives β≈0.91, but the true survival probability is (0.1)^t, while standard walkers decay as (0.6)^t. So restarting is worse, not better, contradicting Table II (which gives x0^crit=1) and Eq. (19) (which claims q>0.78 always helps). The error comes from applying the large-x0 expansion (13) in the small-x0 regime without checking validity.

The reader's concern about the leading-order approximation is valid, but the cancellation issue is more severe: even the exact largest root of (12) is the wrong root when q is largest and canceled. The qualitative existence of a threshold is plausible and likely true, but the correct thresholds must be computed from Q(λ).

This paper is for readers interested in stochastic resetting who want a concrete comparison. As written, it should not be cited or relied upon. It deserves a serious referee because the recurrence and threshold concept are worth examining, and the fix is clear, but only as a major-revision-or-reject situation. I would not cite it in its current form.

Referee Report

1 major / 4 minor

Summary. The paper studies a biased Sisyphus random walker on the nonnegative integers with an absorbing trap at the origin: at each step the walker moves one step toward the trap with probability q and jumps back to its initial position x0 with probability 1-q. The author derives a linear recurrence for the survival probability S(t;x0,q), proposes an exponential decay S(t) ~ alpha beta^t with beta the largest positive root of beta^{x0+1}-beta^{x0}+q^{x0}(1-q)=0, and compares the asymptotic ratio beta with the standard biased-random-walk ratio 2 sqrt(q(1-q)). The paper claims that there is a critical initial gap x0^crit(q), given in closed form in Eq. (17), above which Sisyphus walkers survive longer than standard walkers, and a uniform regime q > about 0.78 where they always survive longer. The recurrence is correctly derived, but the root analysis and the quantitative threshold claims are not.

Significance. The question addressed is genuinely interesting and fits naturally into the restart-and-first-passage literature. The recurrence (9) is derived cleanly, there are no fitted parameters, and Table I provides a useful numerical check of the recurrence. If the root-cancellation issue is repaired, the qualitative conclusion is likely to survive: for fixed q>1/2 the true asymptotic ratio of the Sisyphus walker tends to 1 as x0 grows, whereas the standard-walker ratio 2 sqrt(q(1-q)) is strictly less than 1, so sufficiently large initial gaps should make the Sisyphus walker more persistent. However, as it stands the principal quantitative deliverables—Eq. (17), Table II, and Eq. (19)—are incorrect in important parameter regimes, and the x0 >> 1 approximation is used outside its stated domain. The manuscript needs substantial revision of Sections III and V.

major comments (1)

[§V, Eq. (19)] The uniform claim that for every q > about 0.78 the Sisyphus walker survives longer for all x0 >= 1 is false. A concrete counterexample is q=0.8, x0=1: from the correct reduced polynomial the Sisyphus survival ratio is 1-q=0.2, whereas the standard walker ratio is 2 sqrt(0.8*0.2)=0.8. Thus the Sisyphus walker is much less persistent in this regime. This claim should be removed or replaced by a correct small-x0 analysis; it is not a minor numerical slip but a consequence of the missing root cancellation in Eq. (12).

minor comments (4)

[§III, Eq. (15); §IV, Table I] For x0=10, q=3/4, the approximation (15) gives R_asym=0.9859, while the exact root of Eq. (12) is beta=0.9833 and the numerical ratio in Table I approaches 0.9833. The text says Eq. (15) describes the data 'extremely well'; this should be qualified, and the distinction between the leading-order approximation (15) and the exact root should be stated explicitly.
[Throughout] The phrase 'largest positive root of Eq. (12)' should be replaced by 'largest positive root of the reduced polynomial Q(lambda)' once the cancellation described above is accounted for; otherwise the characteristic equation is overdetermined.
[§II, Eqs. (3)-(5)] The notation Ntot(t) is introduced as a normalized number of walkers, but the text also refers to the 'number' of walkers; it would be clearer to state once that S(t)=Ntot(t) is a survival probability (or fraction of the initial population).
[§V, Table II] The use of the ceiling function is confusing when Eq. (17) gives a negative real threshold, as it does for q=0.9; since x0 is a positive integer, the table should define x0^crit as the smallest integer x0>=1 for which the corrected inequality holds.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the survival recurrence, characteristic-root equation, and critical-gap comparison are derived from the process, not presupposed; the self-cited standard-walker rate is independent; the main concerns are approximation validity, not circularity.

full rationale

The derivation is self-contained. Equation (9) follows from the jump rule (2) and the boundary condition (3), with no fitted parameters; the exponential ansatz (11) is a standard characteristic-root method that leads to the polynomial (12), and the asymptotic ratio (15) is read off from that root. Table I validates the result against the exact recurrence rather than using the claimed answer as an input. The critical-gap formula (17) is obtained by algebraically comparing the Sisyphus ratio (15) with the standard-walker ratio (1), so it is not an input to the derivation. The only self-citation is reference [8] for the standard biased-walker decay rate (1); that rate is a standard, externally verifiable gambler's-ruin result and is not fitted to the present model, so under the stated rules it does not constitute load-bearing circularity. The approximation in (13) and the possible cancellation of the characteristic root beta=q for small x0 are mathematical-validity concerns, not circular reductions: no equation is defined in terms of the claim, and no fitted quantity is relabeled as a prediction. Therefore no significant circularity is present.

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

The derivation is self-contained given the Sisyphus walk model: it uses standard linear-recurrence theory to identify the dominant root β, and a cited standard result for the ordinary walker's decay rate. The only ad hoc element is the uncontrolled leading-order expansion for β, which is used beyond its validity to produce quantitative threshold values. No parameters are fitted to data, and no new entities are introduced.

assumptions (3)

standard math The asymptotic decay rate of a linear homogeneous recurrence is governed by the largest positive root of its characteristic polynomial.
Invoked in Section III when substituting the ansatz S=α β^t into the recurrence (9), producing the polynomial equation (12).
domain assumption The ordinary biased random walk with a trap has asymptotic survival ratio 2√(q(1-q)).
Quoted from refs [8,12] in Eq. (1) and used as the benchmark in Eq. (16).
ad hoc to paper For x0 much larger than 1, the root β is well approximated by 1 - q^{x0}(1-q).
Expansion (13), used to derive Eqs. (14)-(17) and Table II; it is not valid for the small x0 values appearing in the table.

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Cite this review

Pith. "Pith review of Survival probabilities in biased random walks: To restart or not to restart? that is the question." pith.science (2026). https://pith.science/paper/OKLXWLUQ

@misc{pith2026250206667,
  author       = {Pith},
  title        = {Pith review of: Survival probabilities in biased random walks: To restart or not to restart? that is the question},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OKLXWLUQ}},
  note         = {Machine review of arXiv:2502.06667}
}

abstract

The time-dependent survival probability function $S(t;x_0,q)$ of biased Sisyphus random walkers, who at each time step have a finite probability $q$ to step towards an absorbing trap at the origin and a complementary probability $1-q$ to return to their initial position $x_0$, is derived {\it analytically}. In particular, we explicitly prove that the survival probability function of the walkers decays exponentially at asymptotically late times. Interestingly, our analysis reveals the fact that, for a given value $q$ of the biased jumping probability, the survival probability function $S(t;x_0,q)$ is characterized by a {\it critical} (marginal) value $x^{\text{crit}}_0(q)$ of the initial gap between the walkers and the trap, above which the late-time survival probability of the biased Sisyphus random walkers is {\it larger} than the corresponding survival probability of standard random walkers.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

23 extracted references · 19 canonical work pages

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We assume, without loss of generality, that the absorbi ng trap is located at the origin xtrap = 0
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This relation follows from the jumping rule (2), accord ing to which a biased random walker who is located at x0 has to go x0 steps in a row towards the trap (with a characteristic probability q for each step to the left) in order to be absorbed at the origin
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[23]

(11)], where the function β (x0, q ) is determined numerically from the polynomial equation (12)

It is worth noting that the exact ratio Rexact(t) agrees even better with the asymptotic relation S(t + 1; x0, q )/S (t; x0, q ) = β (x0, q ) [see Eq. (11)], where the function β (x0, q ) is determined numerically from the polynomial equation (12). 10

Pith tools

Reviewed August 8, 2026 · model on record in the stance chip above.

[1] [1]

M. N. Barber and B. W. Ninham, Random and Restricted Walks (Gordon and Breach, New York, 1970)

work page 1970

[2] [2]

N. G. van Kampen, Stochastic Processes in Physics and Chemistry (North-Holland, Amster- dam, 1992). 8

work page 1992

[3] [3]

Fernandez, J

R. Fernandez, J. Frohlich, and A. D. Sokal, Random Walks, Critical Phenomena, and Triviality in Quantum Field Theory (Springer Verlag, Berlin, 1992)

work page 1992

[4] [4]

G. H. Weiss, Aspects and Applications of the Random Walk (North Holland, Amsterdam, 1994)

work page 1994

[5] [5]

ben-Avraham and S

D. ben-Avraham and S. Havlin, Diﬀusion and Reactions in Fractals and Disordered Systems (Cambridge University Press, Cambridge, 2000)

work page 2000

[6] [6]

Dickman and D

R. Dickman and D. ben-Avraham, Phys. Rev. E. 64, 020102(R) (2001)

work page 2001

[7] [7]

Marro and R

J. Marro and R. Dickman, Nonequilibrium Phase Transitions in Lattice Models , (Cambridge University Press, Cambridge, 1999); H. Hinrichsen, Adv. Ph ys. 49, 815 (2000)

work page 2000

[8] [8]

Hod, Phys

S. Hod, Phys. Rev. Lett. 90, 128701 (2003) [arXiv:cond-mat/0212055]; S. Hod, Phys. Re v. Lett. 105, 208701 (2010) [arXiv:1009.0941]

arXiv 2003