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Search at bounded speed with immigration

T0 review · 0 major / 3 minor · reviewed 2026-06-29 · grok-4.3

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Pith's one-line read In the fast immigration limit with bounded speed, the kth search time distribution and moments are given exactly by the single searcher's early-time probability distribution.

desk verdict The paper reduces the kth search-time distribution exactly to the single-searcher early-time distribution under fast immigration and bounded speed. read the letter →

arxiv 2605.30629 v1 pith:WZYJROBX submitted 2026-05-28 math.PR cond-mat.stat-mech

Hwai-Ray Tung , Sean D Lawley This is my paper · ORCID
classification math.PRcond-mat.stat-mech
keywords boundedspeedsearchimmigrationtimesprobabilitydistributionstochasticprocessesdiffusionmodelsmomentsoftimefast
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 determines exact expressions for the probability distribution and moments of the kth search time when searchers immigrate quickly but move at finite speed. This matters for biophysical processes where searchers enter progressively, avoiding the unphysical infinite speeds of earlier models. The results are derived rigorously and then applied to several standard search models including variants of diffusion. Comparisons with simulations validate the approach. The work clarifies when the details of individual searcher motion matter for overall search times.

What carries the argument

The exact mapping from the single-searcher early-time probability distribution to the multi-searcher kth search time statistics in the fast immigration limit under bounded speed.

What would settle it

Numerical simulations of the full multi-searcher system with fast immigration and bounded speed that produce a kth search time distribution different from the one calculated from the single searcher's early time probability.

Watch

Extended reading notes

Core claim

We show that in the fast immigration limit the full probability distribution and all the moments of the kth search time can be determined in terms of the early time probability distribution of a single searcher. These mathematical results are applied to canonical models of stochastic search, and different models of diffusion are analyzed to investigate when and how the minutiae of searcher dynamics affect search times.

Load-bearing premise

That the early-time probability distribution of a single searcher is known or computable and that bounded speed permits an exact reduction of the multi-searcher problem to this quantity in the fast immigration limit.

Editorial extensions

If this is right

  • The kth search time can be analyzed using only single searcher early behavior data.
  • Different diffusion models can be compared for their impact on search times.
  • All moments of the search times become explicitly computable.
  • The theory applies directly to biophysical search with immigration.

Reading between the lines

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

  • This suggests that experimental focus on early single-particle trajectories could predict collective search performance in fast-immigration settings.
  • The method might generalize to other speed constraints or time-dependent immigration rates.
  • Optimization of search processes could use these formulas to adjust immigration based on single searcher properties.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 3 minor

Summary. The manuscript studies search times for searchers that immigrate into a bounded domain over time while moving at finite speed. In the fast-immigration limit the authors derive exact expressions for the full probability distribution and all moments of the kth search time, expressed solely in terms of the early-time hitting probability of a single searcher. These formulas are applied to several canonical search models (including different diffusion variants), and the predictions are compared against numerical simulations.

Significance. If the reduction holds, the work supplies a rigorous, parameter-free bridge from single-searcher early-time data to the full multi-searcher search-time statistics under bounded speed. This directly resolves the unphysical infinite-speed artifact that appears in fast-immigration regimes and supplies a practical route to moments and distributions for biophysical search problems. The explicit comparison of distinct diffusion models and the simulation checks are additional strengths.

minor comments (3)
  1. The abstract states that the results are 'rigorous mathematical results,' yet the manuscript should include an explicit statement of the precise technical conditions (domain regularity, speed bound, immigration rate scaling) under which the reduction to the single-searcher early-time distribution is valid.
  2. In the applications section, the comparison between different diffusion models would benefit from a short table summarizing which moments or quantiles differ and by how much, to make the claim that 'minutiae of searcher dynamics affect search times' quantitatively visible.
  3. A brief remark on the numerical method used for the simulations (time-stepping scheme, boundary handling, number of realizations) would strengthen reproducibility of the reported agreement between theory and numerics.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary and recommendation of minor revision. No major comments were provided in the report.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity detected

full rationale

The central result expresses the kth search-time distribution and moments exactly in terms of an independently supplied early-time single-searcher probability distribution under the fast-immigration and bounded-speed assumptions. This is a reduction to an external input quantity rather than a self-definition, fitted parameter renamed as prediction, or self-citation chain. No load-bearing step in the provided abstract or description reduces by construction to the paper's own outputs; the derivation is therefore self-contained against the stated single-searcher benchmark.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

Review based on abstract only; the central assumption of bounded speed is domain-level and the reduction to single-searcher early-time statistics is the key modeling choice. No free parameters or invented entities are mentioned.

assumptions (1)
  • domain assumption Searchers move at bounded (finite) speed
    Explicitly introduced to correct the infinite-speed implication of prior fast-immigration estimates.

how reviews work

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Cite this review

Pith. "Pith review of Search at bounded speed with immigration." pith.science (2026). https://pith.science/paper/WZYJROBX

@misc{pith2026260530629,
  author       = {Pith},
  title        = {Pith review of: Search at bounded speed with immigration},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WZYJROBX}},
  note         = {Machine review of arXiv:2605.30629}
}
abstract

Many biophysical search processes employ searchers which enter or "immigrate" into the domain progressively over time. Existing search time estimates can become unphysical for fast immigration since they imply that searchers move at infinite speed. In this paper, we investigate search times of immigrating searchers that move at bounded speed. In the fast immigration limit, we determine the full probability distribution and all the moments of the $k$th search time in terms of the early time probability distribution of a single searcher. We apply these rigorous mathematical results to several canonical models of stochastic search. We further analyze different models of "diffusion" and use these results to investigate when and how the minutiae of searcher dynamics affect search times. We compare our theory to numerical simulations.

Figures

Figures reproduced from arXiv: 2605.30629 by the authors.

Figure 1
Figure 1. Comparison of theory of section 3 with stochastic simulations of four [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Comparison of the mean fFPTs for the three “diffusion” processes of [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗
Figure 2
Figure 2. Summarizing, this analysis predicts that if [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Extreme First-Passage Time of Many Interacting Particles

    cond-mat.stat-mech 2026-07 conditional novelty 6.0 of 10

    For many interacting Brownian searchers, bounded interactions cannot beat the independent 1/ln N fastest-search time, while coherent pushes and random pairwise kicks can reach 1/N and 1/(N ln N), respectively.

Reference graph

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