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REVIEW 2 major objections 1 minor 39 references

A recursive inverse-transform method generates order statistics of first arrival times from short-time asymptotics without simulating particle paths.

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2026-06-29 23:19 UTC pith:P53A5LDU

load-bearing objection The paper gives a recursive inverse-transform sampler for order statistics of extreme first-passage times that extends from instantaneous to general emission profiles, but the iterative version for slow emission rests on unverified extrapolation of short-time asymptotics. the 2 major comments →

arxiv 2605.25295 v1 pith:P53A5LDU submitted 2026-05-24 math.PR cond-mat.stat-mechq-bio.QM

Accelerated Simulation Algorithms for Extreme First-Passage Problems with General Emission Profiles

Emmanuel Akame Mfoumou , David Holcman This is my paper
classification math.PR cond-mat.stat-mechq-bio.QM
keywords first-passage timesorder statisticsinverse transform samplingdiffusion processessimulation algorithmsextreme value statisticsbounded domainsemission profiles
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper introduces a simulation framework that generates the first k arrival times of diffusing particles to a localized target by sampling directly from asymptotic first-passage distributions. This replaces the need to generate full trajectories for every particle, which becomes prohibitive when particle numbers are large. The approach begins with instantaneous emission and extends iteratively to arbitrary time-dependent emission profiles. It also supplies asymptotic formulas for the mean of the fastest arrival time. The method targets diffusion in bounded domains where short-time asymptotics are known, such as Brownian motion in one, two, or three dimensions.

Core claim

We present a general simulation framework for efficiently generating order statistics of arrival times by exploiting asymptotic first-passage distributions. This framework applies to diffusion processes in bounded domains with localized absorbing targets, for which short-time first-passage asymptotics are available. Starting with the case of instantaneous emission, we derive and implement a recursive inverse transform algorithm to simulate the first k arrivals without tracking particle trajectories. We extend this algorithm to time-dependent emission profiles via an iterative approach, enabling the simulation of extreme statistics in systems with temporal injection, ranging from rapid to pro

What carries the argument

Recursive inverse-transform sampling from the cumulative distribution built on short-time first-passage asymptotics, extended iteratively for time-dependent emission.

Load-bearing premise

The short-time first-passage asymptotics remain accurate enough to serve as the basis for inverse-transform sampling across all emission profiles and order statistics considered.

What would settle it

Apply the algorithm to a one-dimensional interval where exact first-passage distributions are known, generate the first k arrivals, and compare the resulting empirical distribution against the exact order statistics obtained from full Monte Carlo trajectory simulation; significant systematic deviation would falsify the method.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • The algorithm produces the first k arrivals directly from the asymptotic law without generating individual Brownian paths.
  • An iterative extension allows the same procedure to handle arbitrary time-dependent emission profiles.
  • Asymptotic expressions for the mean of the fastest arrival time follow from the same short-time analysis.
  • The resulting samples can be fed directly into models of spatial reaction networks or rare-event detection.

Where Pith is reading between the lines

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

  • The method could be tested for robustness by checking how the generated statistics behave when the target is moved or the domain shape is altered while keeping the same asymptotic form.
  • It opens the possibility of coupling the arrival-time generator to deterministic reaction equations inside the domain without resolving every diffusion path.
  • Accuracy for late-order statistics might degrade if the asymptotic tail does not capture the full distribution, suggesting a hybrid switch to exact sampling once the remaining particles are few.

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

2 major / 1 minor

Summary. The manuscript develops accelerated simulation algorithms for order statistics of extreme first-passage arrival times of diffusing particles to localized absorbing targets. It exploits short-time asymptotic first-passage distributions to derive a recursive inverse-transform sampler for instantaneous emission and extends the method to general time-dependent emission profiles through an iterative integral-equation scheme; asymptotic estimates for the mean fastest arrival time are also supplied.

Significance. If the algorithms are shown to be accurate, the framework could deliver substantial computational gains over full-trajectory Monte Carlo methods when the number of particles is large, with direct relevance to spatial reaction networks and diffusion-controlled activation. The explicit use of known short-time asymptotics for Brownian motion in 1–3 dimensions is a constructive feature.

major comments (2)
  1. [extension to time-dependent emission profiles] The iterative scheme that replaces the constant emission rate with an integral equation whose kernel is built from the short-time tail is introduced without an explicit remainder estimate or convergence analysis. When the emission timescale becomes comparable to or longer than the diffusion timescale, the generated order statistics may leave the regime where the leading asymptotic is uniformly accurate, yet no quantitative control on the resulting error is supplied.
  2. [implementation and validation] The central efficiency and accuracy claims rest on the inverse-transform procedure, yet the manuscript contains no numerical experiments, error tables, or direct comparisons against full-trajectory simulations that would allow assessment of the method’s performance across the claimed range of emission profiles.
minor comments (1)
  1. [preliminaries] Notation for the emission profile and the iterative kernel should be introduced with a single, self-contained definition rather than being redefined piecewise.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive review and the recommendation of major revision. Below we respond point-by-point to the major comments, indicating the changes we will make to the manuscript.

read point-by-point responses

  1. Referee: [extension to time-dependent emission profiles] The iterative scheme that replaces the constant emission rate with an integral equation whose kernel is built from the short-time tail is introduced without an explicit remainder estimate or convergence analysis. When the emission timescale becomes comparable to or longer than the diffusion timescale, the generated order statistics may leave the regime where the leading asymptotic is uniformly accurate, yet no quantitative control on the resulting error is supplied.

    Authors: We thank the referee for highlighting the lack of remainder estimates. The iterative scheme solves the integral equation for the cumulative distribution of arrival times by successive substitution of the leading short-time asymptotic kernel; this is motivated by the fact that the fastest arrivals, which determine the extreme order statistics, occur in the short-time regime where the asymptotic is accurate. We agree that explicit error control would strengthen the presentation. In the revision we will add a dedicated paragraph discussing the truncation error induced by retaining only the leading asymptotic term, together with a heuristic bound derived from the next-order correction in the known first-passage expansion, and we will explicitly delineate the regime (emission timescale much shorter than the typical diffusion time to the target) in which the approximation remains uniformly valid. revision: partial

  2. Referee: [implementation and validation] The central efficiency and accuracy claims rest on the inverse-transform procedure, yet the manuscript contains no numerical experiments, error tables, or direct comparisons against full-trajectory simulations that would allow assessment of the method’s performance across the claimed range of emission profiles.

    Authors: We agree that numerical validation is essential to substantiate the efficiency and accuracy claims. Although the manuscript derives the recursive inverse-transform sampler and the iterative extension in full detail, it does not contain explicit simulation results. In the revised version we will insert a new section (approximately 3–4 pages) that reports Monte Carlo comparisons: (i) instantaneous emission in 1D–3D domains, (ii) several time-dependent emission profiles (constant, exponential, and pulsed), and (iii) tables of relative error in the first k order statistics together with wall-clock time ratios versus full-trajectory Brownian simulations. These experiments will cover the parameter ranges stated in the paper and will quantify both statistical accuracy and computational speedup. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation relies on short-time first-passage asymptotics stated as available from prior literature for Brownian motion in bounded domains (abstract). The recursive inverse-transform algorithm for instantaneous emission and the iterative extension for general emission profiles are constructed on top of these external inputs without any reduction by self-definition, fitted parameters renamed as predictions, or load-bearing self-citation chains visible in the provided text. The central claim of accelerated simulation for order statistics remains independent of the paper's own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no concrete free parameters, axioms, or invented entities; the method presupposes the existence of short-time first-passage asymptotics whose derivation is external to the paper.

pith-pipeline@v0.9.1-grok · 5740 in / 1090 out tokens · 23536 ms · 2026-06-29T23:19:17.122901+00:00 · methodology

0 comments
read the original abstract

Fastest arrival events, where the first among many diffusing particles reaches a target, are central in triggering signal initiation in molecular stochastic systems. Classical approaches to simulate such events rely on full trajectory generation of all particles, leading to prohibitive computational costs in the large particle number regime. In this work, we present a general simulation framework for efficiently generating order statistics of arrival times by exploiting asymptotic first-passage distributions. This framework applies to diffusion processes in bounded domains with localized absorbing targets, for which short-time first-passage asymptotics are available, such as Brownian motion in dimensions one, two, and three. Starting with the case of instantaneous emission, we derive and implement a recursive inverse transform algorithm to simulate the first $k$ arrivals without tracking particle trajectories. We extend this algorithm to time-dependent emission profiles via an iterative approach, enabling the simulation of extreme statistics in systems with temporal injection, ranging from rapid to prolonged emission. Additionally, we provide asymptotic estimates of the mean fastest arrival time. To conclude, the present acceleration algorithm which bypasses Brownian simulations of trajectories can be used for spatial reaction networks, rare event detection, or diffusion-controlled activation.

Figures

Figures reproduced from arXiv: 2605.25295 by David Holcman, Emmanuel Akame Mfoumou.

Figure 1
Figure 1. Figure 1: Substituting Brownian Simulations with Extreme Escape Statistics. This schematic illustrates the core idea of the algorithm: replacing the computationally intensive simulation of Brownian trajectories up to the first escape event with the independent sampling of two random variables — the arrival time τ and the escape target C — from precomputed probability distributions. and set the selected target index … view at source ↗
Figure 2
Figure 2. Figure 2: Iterative Sampling of Successive Escape Events. Recursive procedure used to sim￾ulate the first k = 3 escape events: given the arrival time τk of the k-th escape, τk+1 is sampled conditionally, reflecting the decreasing number of particles in the system. F(t) = 1 − S(t) is given by: (3.6) F(t) ≈    √ 4Dt δ √ π exp  − δ 2 4Dt (1D), √ 2πDt 2 log( 1 ε )δ 2 exp  − δ 2 4Dt (2D), a 2 δ √ πDt exp … view at source ↗
Figure 3
Figure 3. Figure 3: Arrival Times of the k-th Particle : Stochastic Simulation vs Ac￾celerated Algorithm realizations. The markers are averages over simulations and the lines are average over algorithm realizations for the mean (A) and the variance (B) in one dimension. 4. Extreme First-Passage under Uniform Killing Constraints. In this section, we extend our acceleration algorithm to incorporate a uniform killing, where each… view at source ↗
Figure 4
Figure 4. Figure 4: Arrival Time Sampling under Uniform Killing. Each Brownian particle is assigned an exponential lifetime. Candidate escape times generated by the recursive inverse transform method are rejected if they exceed the corresponding killing time. Surviving events are retained and sorted by arrival. first-passage distribution is well-defined. We shall use here for the emission profile a gamma distribution : ϕ(t) =… view at source ↗
Figure 5
Figure 5. Figure 5: Asymptotic behavior of the MFAT in the slow and fast emission regimes vs accelerated simulations. The slow regime is plotted for N = 1000 in A and B in different scales. The asymptotic approximation (eq. 3.2 dotted curve) agrees well with theoretical formula (eq. 2.4) for small α limit. Discrepancies appear as α → 0, due to the approximation of short-time asymptotics for the expression of S(t) beyond their… view at source ↗
Figure 6
Figure 6. Figure 6: Histograms of the first arrival time with killing with 50000 realizations of our simulation algorithm, each of the first arrival time for three killing rates γ ∈ {0, 200, 500}. Killing scenario. To study the impact of a killing rate γ on the arrival times, we generated simulations to collect the histograms of the first arrival time under different values of γ = {0, 200, 500} ( [PITH_FULL_IMAGE:figures/ful… view at source ↗
Figure 7
Figure 7. Figure 7: Comparison between Theoretical and Simulated Mean Arrival Times under Different Emission Regimes. Mean arrival times and their corresponding relative errors between theoretical predictions and accelerated simulations vs number N. (A) Comparing instantaneous emission algorithm: mean arrival times for the 3 rd, 10th, and 20th particles, compared with theoretical values computed from the density of the k th a… view at source ↗

discussion (0)

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Reference graph

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