Elephant Random Walk with multiple extractions

Simone Franchini

REVIEW 4 major objections 3 minor 1 cited by

Elephant Random Walk with multiple extractions

T0 review · 4 major / 3 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This paper establishes that an elephant random walk whose step follows the majority of $k$ odd past draws is exactly a Hill\--Lane\--Sudderth urn with polynomial urn function $\pi_k(y) = (1-p) + (2p-1)P_k(y)$, and that for $k>2$ above a…

desk verdict Proceeding-style re-derivation of the author's own HLS urn results for the k-extraction ERW, with a clean mapping but an unresolved sampling convention and heavy reliance on prior work. read the letter →

arxiv 2507.06478 v1 pith:WQUK2MPD submitted 2025-07-09 math.PR

Simone Franchini This is my paper

classification math.PR MSC 60F1060C05

keywords elephantrandomwalkgeneralizedPolyaurnHLSfunctionlargedeviationsentropydensitysymmetrybreakinglock-in

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 is trying to show that the generalized Elephant Random Walk, where the next step follows the majority of $k$ previously drawn steps with $k$ odd and with probability $p$, has a much richer long-time behavior than the original one-draw model. The author maps the process exactly onto a Hill\--Lane\--Sudderth urn whose urn function is $\pi_k(y) = (1-p) + (2p-1)P_k(y)$, where $P_k(y)$ is the probability that the majority of $k$ draws is positive. Once the mapping is in place, the convergence of the average step $x_N$ is governed by the crossings of $\pi_k$ with the diagonal. For $k>2$ and $p$ above a critical value ($p_c = 5/6$ for $k=3$), the symmetric point $x=0$ becomes unstable and the walk converges almost surely to one of two nonzero limits $x_\pm(p)$; which one is selected depends on the early history. The paper further claims that in the region between the two attractors the entropy density vanishes, so reaching those intermediate values has sub-linear probability cost in $N$.

What carries the argument

The load-bearing object is the Hill\--Lane\--Sudderth (HLS) urn, a two-color urn in which a black ball is added with probability $\pi(y_N)$, where $y_N$ is the current fraction of black balls. The generalized ERW fits this class exactly through the identity $X_n = 2Y_n - 1$, which turns the average step into $y = (1+x)/2$ and the majority probability into the polynomial $P_k(y) = \sum_{h>k/2} \binom{k}{h} y^h(1-y)^{k-h}$; the resulting urn function is $\pi_k(y) = (1-p) + (2p-1)P_k(y)$. Everything else follows from the geometry of $\pi_k$ against the diagonal: stable limits are down-crossings of $\pi_k(y)=y$, unstable equilibria are up-crossings, and the entropy density is obtained from a variational formula, $\phi(y) = -\inf_{\phi\in Q(y)} \int_0^1 d\tau\, L(\partial_\tau\phi(\tau), \pi(\phi(\tau)/\tau))$, with the cumulant generating function satisfying $\partial_\lambda\zeta(\lambda) = \pi^{-1}((e^{\zeta(\lambda)}-1)/(e^\lambda-1))$. The imported large-deviation theorems (Corollaries 5 and 6 of the cited source [18]) supply the zero-entropy interval, the uniqueness of optimal trajectories, and the power-law estimate $O(N^{-(\chi-1)})$ that together produce the phase diagram.

What would settle it

Take $k=3$, $p>5/6$ (for example $p=0.9$), start the walk from two finite histories whose average steps are near $x_+(p)$ and $x_-(p)$, and run many long simulations. The claim predicts that the empirical average stays near the corresponding attractor almost surely, that the chance of ending between $x_-(p)$ and $x_+(p)$ decays as $N^{-(\chi-1)}$ with $\chi=\pi_3'(1/2)$, and that the empirical entropy density $\rho(x)$ is zero inside that interval. A direct estimate showing a strictly negative rate inside the interval, or limiting probabilities for the two basins that do not depend on the initial condition, would contradict Eq. (18).

Watch

Extended reading notes

Core claim

The central claim is Eq. (18): for the generalized ERW with $k>2$, above the critical memory parameter $p_c$ the limit average step satisfies $\lim_{N\to\infty} x_N \in \{x_-(p), x_+(p)\}$ almost surely, where $x_\pm(p)$ are the two stable down-crossings of the urn equation $\pi_k(y)=y$; for $k=3$ these are $x_\pm(p) = \pm\sqrt{(6p-5)/(2p-1)}$ and $p_c=5/6$. Below $p_c$ the urn function crosses the diagonal only at $y_0=1/2$ from top to bottom, and $x_N\to 0$ almost surely, as in the original $k=1$ walk. Above $p_c$ the central point becomes an unstable up-crossing, two symmetric stable fixed points appear, and a fixed initial condition $x_M$ at finite time $M$ selects the nearest basin, with the probability of the farther basin exponentially suppressed in $M$. The accompanying large-deviation statement is that the entropy density $\rho(x) = \lim_{N\to\infty} N^{-1}\log P(x_N = \lfloor xN\rfloor/N)$ is identically zero for $x$ between $x_-(p)$ and $x_+(p)$, and the probability of the whole intermediate interval scales as $O(N^{-(\chi-1)})$ with $\chi$ the derivative of the urn function at the unstable equilibrium.

Load-bearing premise

The whole phase diagram rests on the large-deviation theorems imported from the cited urn literature, and the paper discloses that the same source contains a sign error; if that error reaches the formulas used here, the zero-entropy region and the initial-condition selection would not be established.

Editorial extensions

If this is right

For every odd $k>2$ there is a critical $p_c(k)<1$ above which the walk is bistable: the limiting fraction of positive steps is one of two symmetric nonzero values, and the early history decides which one.
For $k=3$ the two attractors are at $x_\pm(p)=\pm\sqrt{(6p-5)/(2p-1)}$, so the transition happens at $p_c=5/6$.
In the bistable phase the entropy density is zero on the whole interval between the attractors: probabilities of intermediate average steps do not decay exponentially, and the probability of the entire intermediate region is $O(N^{-(\chi-1)})$.
The classic one-draw walk has no such phase: for every $p<1$ the average step converges to zero almost surely, with $p=3/4$ only changing the shape of fluctuations.
The same qualitative picture persists in the infinite-$k$ limit, with a different critical value of $p$.

Reading between the lines

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

The mechanism is probably generic: any symmetric majority rule whose urn function is S-shaped should show the same bistable phase and zero-entropy plateau, so the result extends beyond the binomial $P_k$ to other majority-based reinforcement processes.
Because the selected attractor is fixed by the first $M$ steps, the model offers a minimal stochastic account of history-dependent lock-in: a small early bias can permanently decide between two symmetric outcomes, which may be testable in the market-share setting discussed in the paper.
The disclosed sign error in the source of the large-deviation theorems suggests the quantitative predictions (including the values $p^*=2/3$ and $p^{**}=11/12$ for $k=3$) should be re-derived with corrected signs; if the correction changes the formulas, the phase boundaries could shift.
A finite-$N$ signature of the zero-entropy region is that first-passage times from the unstable center to either attractor should grow more slowly than exponentially; measuring their scaling would provide a direct experimental check independent of the variational calculation.

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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: this is a proceeding-style paper, and the acknowledgements say the paper is a revision of the author's own [19]. The headline results — initial-condition dependence above p_c and the sub-linear entropy region — are imported from [18,19], not proved here. What is genuinely new and useful is the clean statement that the k-extraction ERW with majority rule maps to an HLS urn with polynomial urn function π_k(y)=(1−p)+(2p−1)P_k(y). The k=3 fixed-point analysis, the critical values p_c=5/6, p*=2/3, p**=11/12, and the formula for x±(p) are correct as far as I checked. The paper is transparent about provenance, and the footnote disclosing the sign error in [18] that propagated to [19] and [28] is honest.

Soft spots, in proportion. First, the central claims are not derived in this paper. Eq. (18), the zero-entropy interval (23), and the O(N^{-(χ-1)}) scaling (33) all come from the author's prior work. That is acceptable for a proceeding, but the abstract and intro present them as findings of the paper; a short statement of which results are new versus reviewed would fix the circularity burden. Second, and more importantly, the model definition never says whether the k extractions are with or without replacement. The urn function (12) is the binomial majority probability, exact only for independent draws with replacement. Under the natural without-replacement reading — a uniformly random k-subset of the previous steps — the transition probability is hypergeometric and depends on N, so the HLS mapping is an approximation rather than a theorem. The difference is O(1/N), so the limiting phase diagram probably survives, but a paper built on the exact analogy must specify the convention. Third, the CGF section is mostly speculative ('we expect', 'cannot be integrated'), and the one integrated result relies on a source with a disclosed sign error; it reads as a remark, not a contribution.

Who it's for: readers in the ERW and urn community who want a compact statement of the HLS analogy and the k=3 phase diagram. It is a competent summary, not a new-result paper.

Recommendation: I would send this to a serious referee for a proceedings or a short note, asking them to fix the sampling ambiguity and to add a sentence clarifying which results are new. The math that is done here is sound; the paper just needs to be more careful about what it claims. Desk rejection would be too harsh.

Referee Report

4 major / 3 minor

Summary. The paper defines a generalized elephant random walk in which, at each step, k odd previous signs are drawn and the new sign follows the majority of those k signs with probability p, and opposes it otherwise. For k=1 this reduces to the classical ERW. The author claims that the process can be represented as a Hill-Lane-Sudderth urn with urn function π_k(y) = (1-p) + (2p-1)P_k(y), where P_k(y) is the probability of a positive majority among k independent Bernoulli draws with success probability y. For k=3 he derives a critical value p_c = 5/6: below p_c the limit x_N converges to 0 almost surely, while above p_c the limit is claimed to lie almost surely in {x_-(p), x_+(p)}, with the selected point depending on the initial condition. The paper further claims a region y in [y_-(p), y_+(p)] where the entropy density vanishes, and discusses the cumulant generating function and connections with other HLS-type models. The original analytical content of the present manuscript is mostly the HLS mapping and the fixed-point classification; the large-deviation theorems, the entropy region, and the trajectory formulas are quoted from the author's previous papers [18,19].

Significance. If the claimed results are correct, the paper identifies a simple nonlinear-urn mechanism that produces a phase transition, initial-condition dependence, and a sub-linear entropy region in a natural generalization of the elephant random walk. The mapping for k=3 is elementary and the fixed-point calculation is readily checkable, so the core analogy has some pedagogical and conceptual value. The main limitation is that the central large-deviation statements, the zero-entropy interval, and the O(N^{-(χ-1)}) scaling are not proved in this paper but imported from the author's own [18,19], one of which is acknowledged in a footnote to contain a sign error that propagated to [19] and [28]. The manuscript would be a useful proceedings contribution if the sampling convention is clarified and the status of the quoted theorems is made precise; in its current form, it is not self-contained enough for a full research paper.

major comments (4)

[§1, Eq. (13)] The model definition says the next step is determined by extracting k previous steps from X, but it never states whether the k extractions are with or without replacement. Equation (12) uses the binomial probability P_k(y), which is exact only for independent draws, i.e., sampling with replacement. If the intended process samples without replacement, the probability of a positive majority is hypergeometric and depends on N through n=Ny; it is not a function of y alone, so the process is not exactly an HLS urn with the fixed urn function π_k(y). Since Eqs. (16), (18), and (23) are all derived from π_k, the sampling convention must be stated explicitly, and if without-replacement sampling is intended, the paper must either prove that the HLS analysis still applies or justify the O(1/N) approximation and its effect on the claimed large-deviation rates.
[§3, Eq. (18); §4, Eq. (23)] The central claims -- almost-sure convergence to {x_-(p), x_+(p)}, the zero-entropy interval [y_-(p), y_+(p)], and the O(N^{-(χ-1)}) scaling -- are not proved in this paper; they are imported from Corollaries 5 and 6 of [18] and from [19]. The author's own footnote discloses that Eqs. (2.44) and (2.45) of Corollary 12 of [18] have an inverted sign that propagated to [19] and [28]. Because the present paper gives no independent derivation of the large-deviation results, the reader cannot tell whether the sign error affects the formulas used here. The manuscript should either include the relevant statements and proofs, or state precisely whether the sign error affects (16), (18), and (23) and what the corrected versions imply.
[§3, paragraph after Eq. (18)] The paper asserts that 'it can be shown' that the probability mass near the stable point farther from the initial x_M becomes exponentially suppressed in M, and that choosing M = o(N) concentrates the mass at the closest stable point. This is the basis of the abstract's claim of critical dependence on initial conditions, but no proof or reference is given at that point. Equation (18) itself only asserts that the limit belongs to {x_-, x_+} almost surely, which is compatible with any mixture over the two points. A precise statement with hypotheses on M and the rate of exponential suppression is needed.
[§3, Eq. (16)] As printed, y_±(p) = 1/2 ± sqrt((6p-5)/(2p-1)) gives values outside [0,1] for p > 5/6; for example, p = 0.9 yields y_+ ≈ 1.207, which cannot be a density of black balls. Since Eq. (17) states x_± = ± sqrt((6p-5)/(2p-1)), consistency with y = (1+x)/2 requires y_± = 1/2 ± (1/2) sqrt((6p-5)/(2p-1)). The formula should be corrected and the subsequent statements that depend on it checked.

minor comments (3)

[§1, Eq. (4)] In the definition of ρ(x), the symbol x is used both for the argument of ρ and for the random average x_N inside the probability, which makes the equation ambiguous. Please use two different symbols, e.g., write P(x_N = n/N) and define ρ(x) as the limit of N^{-1} log P(x_N = n/N) with n/N → x.
[§4, Eq. (32)] The equality P(y_N ∈ (y_1,y_2)) = P(y_{⌊τN⌋} ∈ (ψ_1(τ), ψ_2(τ))) is not literally true for an arbitrary fixed τ; it appears to intend equality of probability currents along the optimal trajectories. Please replace this with a rigorous statement or delete the equality, since the subsequent scaling heuristic is already covered by the cited large-deviation results.
[Figure 2 caption] The caption refers to p*, p_c, and p** without defining them in the caption or immediately before the figure. Since these values are used repeatedly in Section 3, please define them where the figure is introduced.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper's derivation is an exact HLS-urn mapping followed by applications of general HLS convergence and large-deviation theorems; the heavy self-citation and the sign-error disclosure are reliability concerns, not definitional circularity.

full rationale

Walking the derivation chain, I find no step where a claimed prediction is equivalent to its input by construction. The central mapping in Sec. 2 is definitional in the harmless sense: Eq. (13) is obtained by substituting the binomial majority probability P_k(y) from Eq. (12) into the conditional probability of a positive step, so the ERW and the HLS urn have the same one-step transition probability. The fixed points in Sec. 3 are obtained by solving the HLS urn equation (14); Eq. (17) follows from the quadratic roots and the change of variable x = 2y - 1. The entropy and CGF statements in Secs. 4-5 are imported from Corollaries 5, 6 and 12 of [18] and from [19], which are mostly the author's own prior work. This is self-citation that is load-bearing in the exposition, but not circular under the reviewing rule: the cited theorems are general statements about HLS urns whose assumptions (continuous, invertible urn functions) do not include the k = 3 target, and the present paper supplies the concrete urn function pi_3. I flag non-circular correctness risks that should be addressed: Eq. (16) appears to omit a factor 1/2 in front of the square root (solving pi_3(y)=y gives y_± = 1/2 ± (1/2) sqrt((6p-5)/(2p-1)), consistent with the x_± in Eq. (17)); the footnote discloses a sign error in [18] that propagated to [19] and [28], weakening the imported large-deviation formulas; and Sec. 1 never states whether the k extractions are with or without replacement, while P_k(y) in Eq. (10) is exact only for with-replacement sampling. These are correctness or exposition issues, not circularity. The derivation's logical structure is input (ERW rule) -> exact urn representation -> cited general theorem -> specialized conclusion, with no equation serving as both premise and conclusion.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper introduces no fitted parameters or invented entities. It rests on imported theorems from the author's prior HLS urn work ([15,16,17,18,19]) and on the monotonicity of the urn function. The main load-bearing axioms are the HLS convergence and large-deviation results, which this paper does not re-derive.

assumptions (6)

domain assumption HLS strong law: the almost sure limit points of y_N are the down-crossings of π(y)=y.
Invoked in Section 3 to assert y_± are stable attractors for p>5/6, citing [15,16,17].
domain assumption Corollary 5 of [18]: for continuous invertible urn functions, φ(y) is strictly convex, negative up to the first crossing, zero between crossings, and negative after the last crossing.
Used in Section 4 to claim the zero-entropy region y∈[y_-,y_+].
domain assumption Corollary 6 of [18]: optimal trajectories are unique and ordered, and P(y_N in an interval) is O(N^{-(χ-1)}) near an unstable equilibrium.
Used in Section 4 to infer logarithmic entropy and initial-condition dependence.
domain assumption The urn function π_k is continuous, strictly increasing (for p>1/2), and maps [0,1] into [0,1], so the HLS variational framework applies.
π_k is a polynomial built from the binomial CDF; stated in Section 2 and used implicitly in Section 4.
standard math Standard martingale convergence theory for reinforced processes.
Implicit in the a.s. convergence statements and the martingale analysis of [18].
domain assumption The fixed-point stability criterion: down-crossings of π(y)=y are stable, up-crossings unstable.
Used throughout Section 3, following Pemantle [17].

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Cite this review

Pith. "Pith review of Elephant Random Walk with multiple extractions." pith.science (2026). https://pith.science/paper/WQUK2MPD

@misc{pith2026250706478,
  author       = {Pith},
  title        = {Pith review of: Elephant Random Walk with multiple extractions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WQUK2MPD}},
  note         = {Machine review of arXiv:2507.06478}
}

abstract

Consider a generalized Elephant Random Walk in which the step is chosen by selecting $k$ previous steps with $k$ odd and then going in the majority direction with a probability $p$ and in the opposite direction otherwise. In the $k=1$ case the model is the original one and could be resolved exactly by analogy with Friedman's urn. However the analogy cannot be extended to the $k>2$ case already. In this paper we show how to treat the model for each $k$ by analogy with the more general urn model of Hill, Lane and Sudderth. Interestingly for $k>2$ we found a critical dependence from the initial conditions beyond a certain values of the memory parameter $p$, and regions of convergence with entropy that is sub-linear in the number of steps.

Figures

Figures reproduced from arXiv: 2507.06478 by the authors.

**Figure 1.** Example of linear urn function π1 (y) for the classic ERW with k = 1, the memory parameter is p ∗ = 3/4. The urn function always crosses the diagonal at y0 = 1/2. [19] critically depends on the initial condition xM, fixed at some finite time M < ∞. We also study the limit entropy density for xN converging to some given x, ρ (x) := lim N→∞ 1 N logP(xN = ⌊xN⌋/N), (4) finding that for k > 2 and above pc there is a regi… view at source ↗

**Figure 2.** Three examples of the the urn function π3 (y) for the generalized ERW with k = 3. The figure shows urn functions for three non-trivial memory parameters, p ∗ = 2/3, pc = 5/6 and p ∗∗ = 11/12. Below pc the urn function down-crosses the line y at y0 = 1/2, that is the only convergence point. For p > pc the point y0 becomes an upcrossing (unstable equilibrium), and the urn function crosses the diagonal y also in y− an… view at source ↗

**Figure 2.** [19] 6 [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

Figures from the paper (2 more)

**Figure 3.** Figure 3: Phase diagram y vs p for the entropy density φ (y) of the classic ERW k = 1, the diagram is shown for p > 1/2 only. For all p < 1 the line at y0 = 1/2 is the only point of convergence for the density of black balls, although there still is a critical p ∗ above which th…

**Figure 4.** Figure 4: Phase diagram x vs p for the entropy density φ (y) of the generalized ERW k = 3. Above the critical value pc = 5/6 the point y0 = 1/2 becomes a unstable equilibrium, and two new symmetric attractors arise according to Eq. (17). In the white colored region we still fin…

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

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