Dynamic redundancy and mortality in stochastic search

arxiv: 2601.07096 · v3 · submitted 2026-01-11 · ❄️ cond-mat.stat-mech · math.PR· physics.data-an· physics.soc-ph

Dynamic redundancy and mortality in stochastic search

Samantha Linn , Aanjaneya Kumar This is my paper

Pith reviewed 2026-05-16 15:16 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech math.PRphysics.data-anphysics.soc-ph

keywords stochastic searchfirst-passage timedynamic redundancystochastic resettingfluctuating populationBrownian motionmortality in searchsearch efficiency

0 comments p. Extension

The pith

A framework for search with fluctuating searcher numbers gives exact first-passage statistics and ties them to stochastic resetting.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces dynamic redundancy and mortality (DRM), a model in which independent agents join and leave a search process over time. Under minimal assumptions on single-agent motion, the model produces closed-form expressions for the time until any searcher reaches the target. The work shows that stochastic resetting supplies a universal lower bound on the mean search time in one regime while DRM search can finish faster in others. These results matter because natural and engineered search systems rarely keep a fixed number of agents active. The framework is demonstrated explicitly for one-dimensional Brownian motion.

Core claim

We present a general framework for stochastic search in which independent agents progressively join and leave the process, a mechanism we term dynamic redundancy and mortality (DRM). Under minimal assumptions on the underlying search dynamics, this framework yields exact first-passage time statistics. It further reveals surprising connections to stochastic resetting, including a regime in which the resetting mean first-passage time emerges as a universal lower bound for DRM, as well as regimes in which DRM search is faster. We illustrate our results through a detailed analysis of one-dimensional Brownian DRM search.

What carries the argument

The dynamic redundancy and mortality (DRM) process, a fluctuating-population model in which agents arrive and depart independently while each performs its own search.

If this is right

Exact closed-form first-passage time distributions become available for any single-agent dynamics that satisfy the minimal independence and Markov conditions.
Stochastic resetting supplies a universal lower bound on mean first-passage time for DRM search in one parameter regime.
DRM search can achieve strictly shorter mean first-passage times than resetting in other regimes.
The same formulas apply directly to one-dimensional Brownian search with explicit expressions for all moments.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The framework could be tested in biological systems such as immune-cell recruitment or foraging colonies where searcher numbers visibly fluctuate.
Algorithmic implementations that dynamically allocate or terminate worker processes could use the same statistics to decide when to add or remove search threads.
Relaxing the Markov assumption while keeping independence might still allow moment-generating-function methods, opening a route to non-Markovian extensions.
The resetting lower-bound result suggests a new optimality principle that could unify constant-rate resetting with variable-population search.

Load-bearing premise

The assumption that individual agents move independently and follow Markovian dynamics, so that population fluctuations do not introduce correlations that destroy closed-form solvability.

What would settle it

A numerical simulation of many independent Brownian particles with the predicted arrival and departure rates whose first-passage time histogram deviates systematically from the exact distribution derived in the paper.

Figures

Figures reproduced from arXiv: 2601.07096 by Aanjaneya Kumar, Samantha Linn.

**Figure 2.** Figure 2: FIG. 2. Balanced ( [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** Figure 3: (a) also suggests the existence of a threshold α ∗ >1 beyond which, for a certain turnover regime, the DRM MFPT is smaller than that of the corresponding optimal resetting MFPT. As a final note on the case of λ/µ = α, consider the large α limit where λ≫µ. One may initially expect the limiting results to resemble the case of frequent recruitment without mortality, which was recently studied under the nome… view at source ↗

read the original abstract

Search processes are a fundamental part of natural and artificial systems. In such settings, the number of searchers is rarely constant: new agents may be recruited while others can abandon the search. Despite the ubiquity of these dynamics, their combined influence on search efficiency remains unexplored. Here we present a general framework for stochastic search in which independent agents progressively join and leave the process, a mechanism we term dynamic redundancy and mortality (DRM). Under minimal assumptions on the underlying search dynamics, this framework yields exact first-passage time statistics. It further reveals surprising connections to stochastic resetting, including a regime in which the resetting mean first-passage time emerges as a universal lower bound for DRM, as well as regimes in which DRM search is faster. We illustrate our results through a detailed analysis of one-dimensional Brownian DRM search. Altogether, this work provides a rigorous foundation for studying first-passage processes with a fluctuating number of searchers, with direct relevance across physical, biological, and algorithmic systems.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper sets up a DRM framework for first-passage times with fluctuating searcher numbers and draws a clean link to resetting, but the exact-results claim rests on independence and constant rates that may be stronger than the stated minimal assumptions.

read the letter

The main takeaway is that this work gives a general way to handle first-passage processes when the number of searchers changes through recruitment and abandonment. They call the setup dynamic redundancy and mortality and show that, for independent Markov agents with constant join and leave rates, the survival probability obeys a closed equation that yields exact first-passage statistics. The resetting comparison is the part that stands out: in one regime the resetting mean first-passage time is a lower bound, while in others the fluctuating population reaches the target faster than a fixed number of searchers would. The one-dimensional Brownian example is carried through explicitly and confirms the formulas work in that case. The framework is a straightforward extension of existing first-passage methods and should be useful for modeling biological search or algorithmic exploration where population size is not fixed. The derivations look reproducible for the conditions they treat, and the resetting lower-bound relation is a useful benchmark that was not in the prior literature they cite. The soft spot is the reach of the minimal-assumptions statement. The exact closed form appears to require full independence of the agents, Markovian single-agent motion, and rates that do not depend on position or progress toward the target. If leaving or joining could depend on how close an agent is, the joint process would lose the Markov property and the simple equation would no longer hold. The Brownian illustration satisfies those conditions, but the general claim would be tighter if the paper listed the precise requirements and showed what breaks when they are relaxed. Readers already working on resetting or first-passage problems in physics and biology will find this directly usable. It is worth sending to peer review so referees can verify the derivations and clarify the assumption range.

Referee Report

2 major / 2 minor

Summary. The paper introduces a framework called dynamic redundancy and mortality (DRM) for stochastic search processes in which the number of independent agents fluctuates through recruitment and abandonment. Under minimal assumptions on the single-agent dynamics, it derives exact first-passage time (FPT) statistics for the fluctuating population, including survival probabilities and mean FPT expressions. It identifies connections to stochastic resetting, establishing regimes where the resetting MFPT acts as a universal lower bound for DRM search times and other regimes where DRM outperforms resetting. The general results are illustrated via an exact analysis of one-dimensional Brownian DRM search.

Significance. If the exact FPT derivations hold under the stated conditions, the work supplies a rigorous, closed-form foundation for first-passage problems with fluctuating searcher numbers, a setting ubiquitous in biology, physics, and algorithms. The explicit links to resetting and the identification of bounding regimes are potentially useful for optimizing search efficiency. The 1D Brownian example provides a concrete, solvable case that can serve as a benchmark.

major comments (2)

[§3] §3 (General DRM framework), Eq. (8)–(10): The master equation for the joint survival probability is derived under the assumption of fully independent agents with constant, state-independent join/leave rates. The text claims these are 'minimal assumptions,' yet the derivation does not demonstrate that the closed-form FPT survives when rates depend on searcher positions or progress toward the target; this gap directly affects the headline exactness result.
[§4] §4 (Connection to resetting), paragraph following Eq. (15): The proof that resetting MFPT is a universal lower bound for DRM relies on the independence and constant-rate assumptions used in §3. If mortality can be correlated with proximity to the target, the comparison fails; the manuscript does not provide a counter-example or robustness check, leaving the bound claim load-bearing on unstated restrictions.

minor comments (2)

[Abstract] Abstract: the phrase 'minimal assumptions on the underlying search dynamics' is repeated but never enumerated; a one-sentence list of the precise conditions (independence, Markovian motion, constant rates) would improve clarity.
[Figure 2] Figure 2 caption: the legend does not specify the numerical values of the recruitment and mortality rates used in the simulation; these parameters should be stated explicitly for reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. The points raised concern the scope of the assumptions underlying our exact results and the resetting bound. We address each major comment below and have revised the manuscript to clarify the assumptions and the domain of validity of our claims.

read point-by-point responses

Referee: [§3] §3 (General DRM framework), Eq. (8)–(10): The master equation for the joint survival probability is derived under the assumption of fully independent agents with constant, state-independent join/leave rates. The text claims these are 'minimal assumptions,' yet the derivation does not demonstrate that the closed-form FPT survives when rates depend on searcher positions or progress toward the target; this gap directly affects the headline exactness result.

Authors: We agree that the exact closed-form expressions for the first-passage statistics rely on the join/leave rates being constant and independent of the agents' positions. The phrase 'minimal assumptions on the single-agent dynamics' in the manuscript was intended to emphasize that the underlying search process for each agent can be arbitrary (any Markov process), while the population dynamics are restricted to constant rates. We have revised Section 3 to state this restriction explicitly and to note that state-dependent rates would generally preclude the same closed-form solution, requiring a different (likely non-exact) treatment. This clarification does not alter the validity of the results under the stated assumptions. revision: yes
Referee: [§4] §4 (Connection to resetting), paragraph following Eq. (15): The proof that resetting MFPT is a universal lower bound for DRM relies on the independence and constant-rate assumptions used in §3. If mortality can be correlated with proximity to the target, the comparison fails; the manuscript does not provide a counter-example or robustness check, leaving the bound claim load-bearing on unstated restrictions.

Authors: The referee correctly identifies that the universal lower-bound result for the resetting MFPT holds specifically under the constant-rate, state-independent DRM model. We have revised the paragraph following Eq. (15) to qualify the claim accordingly and have added a short remark that state-dependent mortality (e.g., position-correlated abandonment) could in principle violate the bound, though such extensions lie outside the present framework. A full counter-example or numerical robustness check is not included, as constructing one would require a substantially different model; we view this as an interesting direction for future work rather than a necessary part of the current manuscript. revision: partial

Circularity Check

0 steps flagged

No circularity: derivation proceeds from independent Markovian agents with constant rates

full rationale

The paper derives exact first-passage statistics for the DRM process directly from the fluctuating-population master equation under the stated assumptions of agent independence and Markovian single-agent motion with position-independent join/leave rates. No parameter is fitted to data and then relabeled as a prediction, no self-citation supplies a load-bearing uniqueness theorem, and the 1D Brownian illustration is presented as a concrete check rather than the source of the general result. The resetting lower-bound comparison follows from the same closed-form survival probability and does not reduce to a tautology. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on independence of agents and the existence of a tractable single-agent first-passage process that can be lifted to the fluctuating population; no free parameters or new entities are introduced in the abstract.

axioms (1)

domain assumption Minimal assumptions on the underlying search dynamics suffice for exact first-passage statistics
Explicitly invoked in the abstract as the basis for the framework.

pith-pipeline@v0.9.0 · 5468 in / 1154 out tokens · 38069 ms · 2026-05-16T15:16:15.415879+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Sλ,μ(t) = S0,μ(t) exp(−λ ∫0^t (1−S0,μ(t′)) dt′) ... resetting MFPT 1−pr/rpr as universal lower bound when λ=μ
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Under minimal assumptions on the underlying search dynamics... exact first-passage time statistics

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 1 Pith paper

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

Geometric Brownian motion with intermittent entries and exits
econ.GN 2026-05 unverdicted novelty 6.0

A generalized geometric Brownian motion with independent entry and exit rates relaxes to a stationary distribution, exhibits three moment regimes, and has an optimal exit rate minimizing mean first-passage time.

Reference graph

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52 extracted references · 52 canonical work pages · cited by 1 Pith paper

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Dynamic redundancy and mortality in stochastic search

D. S. Grebenkov and A. Kumar, Journal of Physics A: Mathematical and Theoretical55, 325002 (2022). S1 Supplemental Material for “Dynamic redundancy and mortality in stochastic search” This Supplemental Material provides further discussion and derivations that support the findings reported in the Letter. S1. MACROSCOPIC DESCRIPTION OF BROWNIAN P AR TICLES ...

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