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REVIEW 3 major objections 3 minor 48 references

Extreme First-Passage Time of Many Interacting Particles

T0 review · 3 major / 3 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read A no-go theorem proves that bounded interactions cannot accelerate the earliest hit of N Brownian searchers below the 1/ln N barrier, with two sharp exceptions that break it.

desk verdict First general framework for extreme first-passage with interactions, but all core proofs sit in an unavailable SM; the reader's central objection doesn't survive contact with the sign of the repulsive force. read the letter →

arxiv 2607.22528 v1 pith:27OGSA5D submitted 2026-07-24 cond-mat.stat-mech cond-mat.mes-hallcond-mat.softmath-phmath.MPphysics.bio-ph

Ruicheng Bao This is my paper · ORCID
classification cond-mat.stat-mechcond-mat.mes-hallcond-mat.softmath-phmath.MPphysics.bio-ph MSC 60J6560G7082C22 PACS 05.40.Fb05.10.Gg
keywords extremefirst-passagetimeinteractingBrowniansearchersno-gotheoremlogarithmictimescalerepulsiveinteractionscoherenttransportstochasticforcingmany-bodysearch
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 asks whether interactions among many searchers can make the first successful arrival—the extreme first-passage time—faster than the classic 1/ln N scaling of N independent Brownian searchers. It answers with a no-go theorem: for broad classes of bounded forces, interactions cannot beat the logarithmic barrier, and for repulsive interactions this barrier is exact at leading order. Two mechanisms do break the barrier: deterministic coherent push can reduce the time to order 1/N, and stochastic pairwise kicks that amplify effective diffusivity reach 1/(N ln N). A unified acceleration limit interpolates between these regimes, cleanly separating acceleration due to statistical redundancy from that generated by coherent many-body transport or amplified fluctuations. This framework matters because most real search systems, from cellular signaling to foraging, involve interactions that break the independent-particle assumption.

What carries the argument

The proof machinery projects each searcher's motion onto a fixed targetward direction and treats it as a one-dimensional Brownian motion with a drift budget B_N; the reflection principle plus a union bound then convert short-time drift control into a lower bound on the earliest-hit probability. The matching upper bound uses a comparison inequality, P(T_N>t) ≤ ∏ erf(z_i(0)/√(4Dt)), valid for reciprocal purely repulsive interactions or confined systems, which proves that repulsion cannot delay the extreme first-passage relative to independent searchers. The two escape mechanisms are explicit many-body constructions: a screened single-file channel where the leader feels an O(N) coherent push fr

What would settle it

Take two searchers on the half-line with constant, unbounded-range repulsive force between them, a planar absorbing target at 0, and initial positions ℓ0 and ℓ0+δ. Compute the joint survival probability numerically: if it exceeds the product erf(ℓ0/√(4Dt))·erf((ℓ0+δ)/√(4Dt)) at any time t, then Eq. (10) fails and the exact logarithmic class is not established. Alternatively, in a many-body simulation with bounded finite-range repulsion but a superlogarithmic number of neighbors per particle, observe whether E[T_N] decays faster than 1/ln N; if it does not, the no-go theorem's domain is wider t

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Extended reading notes

Core claim

The central claim is a no-go theorem plus an exact logarithmic class. For any interacting overdamped system in which, up to the first hit, the targetward drift accumulated by each searcher is o(ln N) on the timescale t = c/ln N, the mean extreme first-passage time satisfies liminf (ln N) E[T_N] ≥ ℓ0^2/(4D). When a growing number of searchers start within a fixed distance of a flat target and are either confined in a slab or repel each other reciprocally, a matching upper bound gives equality, so 1/ln N is exact for these classes. The two exceptions are explicit: coherent force accumulation (screened single-file repulsion) gives 1/N, and stochastic pair kicks with enhanced diffusivity give 1/

Load-bearing premise

The claim that repulsive interactions cannot delay the extreme first-passage time relative to independent searchers—the inequality P(T_N>t) ≤ ∏ erf(zi(0)/√(4Dt))—is assumed without a range or decay condition on the repulsion, and its proof is deferred to the supplement; a long-range repulsion that pushes a nearby searcher away from the target while pushing a distant one toward it could make the inequality fail.

Editorial extensions

If this is right

  • For soft, finite-range, or other bounded-force interactions with logarithmically bounded initial local density, the classic 1/ln N search time is the universal fastest scale; interactions alone cannot improve it.
  • Under purely repulsive reciprocal interactions or with confinement, the asymptotic constant ℓ0^2/(4D) is exact: statistical redundancy, not interaction strength, sets the leading speed of the earliest arrival.
  • To beat the logarithmic barrier at fixed initial gap, a system must either generate coherent targetward transport (deterministic, giving 1/N^p for (p+1)-body forces) or amplify fluctuations (stochastic, giving 1/(N^q ln N)).
  • The unified lower bound means any faster search must consume a budget of drift or diffusivity on the short-time window; the two explicit models attain the pairwise limits, so those branches are optimal.
  • The results separate the three sources of speedup—redundancy (logarithmic), coherent many-body transport (algebraic), and amplified fluctuations (algebraic times logarithmic)—providing a classification for interacting extreme-statistics problems.

Reading between the lines

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

  • A direct corollary the author leaves unexplored: the same no-go logic should apply to bounded-speed, non-Gaussian, or time-correlated noise whenever short-time tails are exponential; testing this would generalize the logarithmic class beyond Brownian motion.
  • The comparison inequality suggests a practical diagnostic: measure the ratio of interacting to independent many-body survival probability; any excursion above 1 would mark the onset of interaction-driven slowdown, and the paper's O(ln N)-neighbor threshold gives a concrete design rule for experiments.
  • For biological search, the 1/N coherent-push branch implies that collectively pushing one leader is the most efficient route to speed up first arrival; quorum-sensing or alignment interactions that produce a global drift could approach this limit, whereas purely local repulsion cannot.
  • The unified bound could be turned into an optimization principle: among all symmetric interactions with a fixed interaction budget, the fastest extreme search is achieved either by concentrating force on one target-facing coordinate or by injecting zero-mean noise that raises effective diffusivity; mixtures of the two mechanisms interpolate continuously between the limits.
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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

3 major / 3 minor

Summary. The paper studies the extreme first-passage time T_N of N interacting overdamped Brownian searchers toward an absorbing target. It states three main results: (i) a no-go theorem (Theorem 1) showing that bounded force budgets cannot beat the 1/ln N timescale of independent searchers, with a corollary for bounded finite-range repulsion under initial coordination control; (ii) matching logarithmic upper bounds (Proposition 1), including the strong comparison inequality P(T_N>t) ≤ ∏_i Q0_i(t) for reciprocal purely repulsive interactions; and (iii) a unified acceleration limit (Theorem 2) yielding N^{-p} and (N^q ln N)^{-1} speed limits for deterministic (p+1)-body and stochastic (q+1)-body forcing, together with two explicit solvable models attaining the pairwise cases. The paper also discusses mechanisms that escape the logarithmic class and presents the claimed results as establishing an exact logarithmic universality class for generic bounded interactions.

Significance. If the statements are correct, the paper would establish a sharp and surprisingly general universality result: many bounded interactions cannot change the logarithmic extreme-search timescale, while coherent many-body forces and enhanced fluctuations provide algebraically faster but tightly bounded acceleration. The results are parameter-free and carry explicit constants (e.g., ℓ0^2/(4D) in Eq. (6), ℓ0^2/(4κ) in Eq. (14)), so they make falsifiable predictions. The two solvable examples and the interpolation theorem provide a clear physical taxonomy of interaction-driven speedup. The main weakness is not internal inconsistency but verifiability: almost all proofs are deferred to a Supplemental Material that is not part of the submitted text, and the paper's central distributional comparison in Eq. (10) is asserted without a proof in the main text.

major comments (3)
  1. [Theorems 1, Proposition 1, Theorem 2, Example A (SM [41])] All load-bearing derivations are deferred to the Supplemental Material [41], which is not included. This includes the proof of Theorem 1 (SM S2 A–B), Proposition 1 (SM S3 A–F), the sharpness constructions for Theorem 2 (SM S5 D–E), and the mean-time estimate for Example A in Eq. (12). Without the SM, I cannot verify the central claims or the claimed optimality of the scaling exponents. The manuscript must either include the SM or present enough of these proofs in the main text for independent checking.
  2. [Proposition 1(ii), Eq. (10)] Eq. (10) is a very strong statement: for arbitrary-strength reciprocal purely repulsive interactions, the interacting survival probability is bounded above by the product of independent single-particle survivals. This comparison is the load-bearing step for the exact logarithmic upper bound, but the main text gives no proof, only an intuitive remark. The reader's worry that a near particle could be pushed away is a sign error—the minimal-z particle is pushed toward the target by all particles above it—so the physical scenario is plausible. However, a rigorous proof of the label-switching/minimum-process comparison is still required; this is not a minor omission because it is the core of the upper-bound claim.
  3. [Table I and End Matter C] The optimality of the two branches in Table I (Example A attaining N^{-1}, Example B attaining (N ln N)^{-1}) is established only by deferred SM constructions. The main text states that a smooth tagged-leader construction 'exactly attains this scale,' but the construction and its verification are in SM Sec. S5 E. Similarly, Eq. (12) for Example A is justified by 'the argument detailed in [41].' These sharpness claims are essential to the paper's message that the bounds in Theorem 2 are optimal, and they cannot be checked from the current text.
minor comments (3)
  1. [Eq. (15)] The formula in Eq. (15) appears dimensionally consistent with the threshold derived from Eq. (25), so I do not see a dimensional error; however, the formula should be checked in the final typeset version because the radical scope in the plain text is ambiguous.
  2. [References and footnotes] Reference [41] should be supplied with a URL or included as an attachment. Footnote [46] refers to 'the WCA interaction' without defining the acronym; please spell it out. The 'numerical demonstration' in SM Sec. S8 is not shown; if it supports a claim, include the figure or a reproducible description.
  3. [Notation, Eq. (3)] The definition of ℓ0 as an infimum over N and i is clear, but the subsequent use of ℓ0 in Theorem 1 as a universal lower bound is not explicitly restated; a small reminder near Theorem 1 would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivation chain is self-contained and no prediction reduces to a fitted input or imported self-citation.

full rationale

The paper's central results are derived from the stated Langevin model and explicit short-time budgets, not from the conclusions they target. Theorem 1 is a union-bound/reflection-principle estimate on stopped projected displacements (Eq. 4), Proposition 1 is a stated comparison inequality (Eq. 10) whose proof is deferred to the Supplemental Material but which is not assumed in the no-go theorem or in the acceleration limit, and Theorem 2 follows algebraically from the threshold equation B_N t + 2 sqrt(D_eff t ln N) = l0 (End Matter C, Eq. 25). Examples A and B are solvable model systems that realize the derived bounds; no parameter is fitted to force the claimed 1/N or 1/(N ln N) scalings. The only self-references are the paper's own Supplemental Material [41] for proof details, which is a deferral of verification rather than an imported conclusion, and one incidental self-citation [6] in the introduction. The manuscript openly flags limitations such as unproved singular-core cases, nonreciprocal interactions, and AI assistance in proofs, but none of these indicate that any result is equivalent by construction to its inputs. The reader's concern about Proposition 1(ii) is a potential correctness matter for the comparison inequality, not a circularity, because the inequality is not derived from the logarithmic timescale it is used to prove.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The results are theorem-based; the only imported inputs are standard stochastic-calculus facts and the overdamped Langevin model. The main unassessed premise is Proposition 1(ii)'s product comparison, whose proof is deferred to the SM and whose statement lacks range/decay conditions on repulsive interactions.

assumptions (5)
  • standard math The one-dimensional reflection principle and union bound control the hitting probability of a projected Brownian motion with drift budget.
    Used in Theorem 1 and Theorem 2 to obtain erfc tails; standard but unproved in the text.
  • standard math Dambis-Dubins-Schwarz time-change for continuous local martingales reduces projected martingales to Brownian motion.
    Invoked in End Matter C and SM S6A; standard stochastic calculus fact.
  • domain assumption Searchers follow the overdamped Langevin equation with independent Brownian motions and absorbing target.
    Eq. (1)-(2); all results are conditional on this dynamics.
  • domain assumption The initial gap ell0 is bounded away from zero independently of N.
    Eq. (3); the entire logarithmic benchmark depends on this fixed initial distance.
  • ad hoc to paper For reciprocal purely repulsive interactions, P(T_N > t) <= prod_i Q0_i(t) even with no restriction on interaction strength or range.
    Proposition 1(ii), Eq. (10); this comparison is load-bearing for the logarithmic upper bound and is unverified in the main text. No range/decay condition is stated, and a constant long-range repulsion can push the nearest searcher away from the target, potentially violating the inequality.

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Cite this review

Pith. "Pith review of Extreme First-Passage Time of Many Interacting Particles." pith.science (2026). https://pith.science/paper/27OGSA5D

@misc{pith2026260722528,
  author       = {Pith},
  title        = {Pith review of: Extreme First-Passage Time of Many Interacting Particles},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/27OGSA5D}},
  note         = {Machine review of arXiv:2607.22528}
}
abstract

Extreme first-passage events are broadly relevant to biological, chemical, and physical processes in which the first successful arrival determines the outcome. Existing theories are confined to noninteracting searchers. Interacting extreme-statistics problems are notoriously difficult because correlations destroy probability factorization. We establish a general framework for interacting extreme search. A no-go theorem shows that broad classes of bounded interactions cannot beat the $1/\ln N$ extreme timescale of $N$ independent Brownian searchers, and complementary upper bounds prove that this scale is exact for broad classes of repulsive interactions. We then identify two sharp mechanisms beyond the logarithmic class and derive a unified interaction-driven acceleration limit. In particular, deterministic pairwise interaction can at most reduce the extreme search time to order $1/N$, while stochastic pairwise forcing attains $1/(N\ln N)$. Our results separate acceleration due to statistical redundancy from that generated by coherent many-body transport or amplified fluctuations, deepening our understanding of interacting stochastic systems.

Figures

Figures reproduced from arXiv: 2607.22528 by the authors.

Figure 1
Figure 1. FIG. 1. Two many-body acceleration mechanisms at fixed [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗

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