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arxiv: 2604.10004 · v1 · submitted 2026-04-11 · ❄️ cond-mat.stat-mech · math.PR

Universality and ambiguity in extremes of anomalous diffusion

Sean D Lawley This is my paper

Pith reviewed 2026-05-10 16:26 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech math.PR
keywords fastest first passage timeanomalous diffusionfractional Brownian motionextreme value statisticssubdiffusionsearch processesbiophysical modelingbounded speed
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The pith

In models of normal and anomalous diffusion the fastest search time among many particles shrinks only logarithmically with more particles, and subdiffusion can beat normal diffusion.

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

The paper examines the fastest first passage time, the moment when the quickest of many random searchers reaches a target. Across scaled Brownian motion, Riemann-Liouville fractional Brownian motion, and fractional Brownian motion, with either bounded or unbounded speeds, this time decreases logarithmically as the number of searchers grows. In appropriate parameter ranges subdiffusive motion reaches the target sooner than ordinary diffusion, while superdiffusive motion reaches it later. These two features therefore appear across the whole class of models examined. The precise ranges of parameters where the features hold, however, depend on the details of each individual process.

Core claim

For scaled Brownian motion, Riemann-Liouville fractional Brownian motion, and fractional Brownian motion the fastest first passage time decays logarithmically in the number of searchers. Subdiffusion can be faster than normal diffusion and superdiffusion can be slower. These behaviors are universal across the models, yet the parameter regimes in which they appear are model-dependent.

What carries the argument

Fastest first passage time (fFPT) extracted from the extreme-value asymptotics of the first-passage distributions of the chosen diffusion processes.

If this is right

  • The fastest search time vanishes only logarithmically with growing searcher number in all the models considered.
  • Subdiffusive search can reach a target faster than normal diffusion inside certain parameter windows.
  • Superdiffusive search can reach a target more slowly than normal diffusion inside certain parameter windows.
  • The logarithmic decay and the relative speed ordering hold for both bounded-speed and unbounded-speed versions of the processes.
  • The intervals of parameters where these statements are valid differ from one diffusion model to another.

Where Pith is reading between the lines

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

  • If experiments with real particles that have bounded speed still display logarithmic decay, then bounded speed alone does not force exponential convergence to a positive minimum time.
  • Search efficiency in crowded biological settings may improve only modestly once the number of searchers exceeds a few dozen.
  • The same logarithmic scaling may appear in other processes whose first-passage tails admit extreme-value asymptotics, even if they are not fractional Brownian motions.

Load-bearing premise

The covariance structures and scaling properties of the diffusion processes must allow extreme-value asymptotics to describe the tail of the first-passage distribution.

What would settle it

A direct measurement showing that the minimal arrival time among many subdiffusive particles falls exponentially, rather than logarithmically, with searcher number would contradict the claimed universality.

Figures

Figures reproduced from arXiv: 2604.10004 by Sean D Lawley.

Figure 1
Figure 1. Figure 1: FIG. 1. Example trajectories of a bounded speed anomalous [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Convergence in distribution of fFPTs of the unbounded speed sBm in (13) and its bounded speed analog in (36). [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Decay of the mean fFPT as [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. The convergence in distribution in (52) for [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Finite speed RLfBm in (40) agrees with the un [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. The convergence in distribution in (57) for [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
read the original abstract

Many biophysical processes begin when the fastest searcher finds a target out of many random searchers, which is called an extreme or fastest first passage time (fFPT). In some models, (i) the fFPT vanishes logarithmically as the number of searchers grows, and (ii) the fFPT can be faster for subdiffusive search compared to normal diffusion. Though mathematically rigorous, the relevance of (i) and (ii) to actual physical systems is suspect since their derivations involve searchers which move with unbounded speed. Indeed, we previously proved that the fFPT for searchers with bounded speed converges exponentially to a strictly positive minimal search time as the number of searchers grows. In this paper, we study fFPTs for a broad class of anomalous and normal diffusion models with bounded or unbounded speed. These models include scaled Brownian motion, Riemann-Liouville fractional Brownian motion, and fractional Brownian motion. For all of these models, we show that the fFPT decays logarithmically in the number of searchers and that subdiffusion can be faster than normal diffusion (we further show that superdiffusion can be slower than normal diffusion). In this sense, features (i) and (ii) are rather universal. On the other hand, we show that the parameter regimes in which (i) and (ii) are valid depend on the particulars of the individual model, and thus ambiguities remain in the relevance of these features to specific physical 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.

Referee Report

2 major / 1 minor

Summary. The manuscript studies the fastest first-passage time (fFPT) for N searchers in a class of anomalous and normal diffusion processes (scaled Brownian motion, Riemann-Liouville fractional Brownian motion, and fractional Brownian motion) that admit both bounded- and unbounded-speed realizations. It asserts that, for all these models, the fFPT decays logarithmically in N and that subdiffusive search can be faster than normal diffusion (with the converse for superdiffusion), rendering these features universal, while the admissible parameter regimes remain model-dependent and thus introduce ambiguities for physical interpretation.

Significance. The paper supplies rigorous proofs for the logarithmic decay and the speed comparisons across the listed models. If these derivations hold, the results indicate that the logarithmic vanishing of the fFPT and the counter-intuitive ordering of anomalous versus normal diffusion in extreme statistics are more general than earlier bounded-speed analyses suggested, extending the scope of such phenomena while still flagging model-specific limitations on applicability.

major comments (2)
  1. [Abstract] Abstract: the manuscript recalls its authors' prior proof that bounded-speed searchers yield fFPT converging exponentially to a strictly positive minimal time, yet claims logarithmic decay 'for all of these models' including those with bounded speed. The text must explicitly reconcile this by showing either that the bounded-speed versions of scaled BM, RL-fBM and fBM lack a positive lower bound on first-passage time or that their tail asymptotics evade the earlier exponential result; this distinction is load-bearing for the universality statement.
  2. [Main derivations (unspecified section)] The derivations rest on the specific covariance structures and scaling properties of the three processes together with extreme-value asymptotics applied to their first-passage distributions. Explicit conditions, error bounds, or handling of boundary cases under which these asymptotics remain valid (especially when speed is bounded) should be stated to substantiate the claimed rigor.
minor comments (1)
  1. Notation for the covariance functions and the precise definitions of bounded-speed realizations should be introduced with a short table or explicit formulas to aid readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and valuable comments that highlight key points for improving the clarity and precision of our claims. We address each major comment below and will make the necessary revisions to the manuscript.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the manuscript recalls its authors' prior proof that bounded-speed searchers yield fFPT converging exponentially to a strictly positive minimal time, yet claims logarithmic decay 'for all of these models' including those with bounded speed. The text must explicitly reconcile this by showing either that the bounded-speed versions of scaled BM, RL-fBM and fBM lack a positive lower bound on first-passage time or that their tail asymptotics evade the earlier exponential result; this distinction is load-bearing for the universality statement.

    Authors: We agree that the abstract requires clarification to avoid any apparent contradiction with our prior results on bounded-speed searchers. In the revised version, we will explicitly state that the logarithmic decay is established for the unbounded-speed realizations of scaled Brownian motion, Riemann-Liouville fractional Brownian motion, and fractional Brownian motion. For the bounded-speed realizations, we will note that these models can be constructed such that the first-passage time distributions lack a strictly positive lower bound independent of path (due to the specific scaling and covariance properties allowing arbitrarily small passage times with positive probability in the relevant parameter regimes), thereby evading the exponential convergence to a minimal time. This distinction will be added to the abstract and elaborated in the introduction, preserving the universality statement while making the model-dependent applicability transparent. revision: yes

  2. Referee: [Main derivations (unspecified section)] The derivations rest on the specific covariance structures and scaling properties of the three processes together with extreme-value asymptotics applied to their first-passage distributions. Explicit conditions, error bounds, or handling of boundary cases under which these asymptotics remain valid (especially when speed is bounded) should be stated to substantiate the claimed rigor.

    Authors: We acknowledge the need for greater explicitness in the conditions of validity. In the revised manuscript, we will add a new subsection detailing the precise parameter regimes (e.g., ranges of the scaling exponent for scaled Brownian motion and the Hurst parameter for the fractional processes) under which the extreme-value asymptotics for the first-passage time tails hold. We will include error bounds obtained from the application of extreme-value theory to the relevant distributions and discuss boundary cases, including bounded-speed realizations, where the asymptotics may cross over to the exponential behavior established in our earlier work. These additions will be supported by expanded proofs in the appendix to substantiate the rigor of the derivations. revision: yes

Circularity Check

0 steps flagged

Minor self-citation to prior bounded-speed result; central universality claims derived independently for new models

full rationale

The paper references its own prior proof that bounded-speed searchers have fFPT converging exponentially to a positive minimum, but this is used only to motivate the study of anomalous models. The new results for scaled BM, RL-fBM, and fBM rely on their specific covariance structures and extreme-value asymptotics, which are distinct from the prior work. No derivation step reduces to a fitted parameter or self-referential definition, making the self-citation non-load-bearing.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on established properties of stochastic processes without introducing new free parameters or entities.

axioms (2)
  • standard math Standard Gaussian increment and covariance properties of Brownian motion and fractional Brownian motion.
    Used to define the diffusion models and derive first-passage asymptotics.
  • domain assumption Extreme-value theory applies to the tail behavior of first-passage times for these processes.
    Invoked to obtain the logarithmic decay in the many-searcher limit.

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Works this paper leans on

47 extracted references · 47 canonical work pages · 2 internal anchors

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    or superdiffusion (α∈(1,2)) for sufficiently largeN. E. More detailed fFPT asymptotics Roughly speaking, (19) means that P(τ≤t)≈At pe−L2/(2σ2tα) if 2σ 2tα/L2 ≪1, for some constantsA >0 andp∈R. We are not aware of the values ofAandpfor RLfBm or fBm. However, ifτ Bm denotes the FPT in (18) for the normal diffusion case ofα= 1, then it is well-known that P(τ...

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    The mean-squared displacement ofX ε in (40) has been determined exactly in terms of the hypergeometric func- tion [23], and at large time, it is given by [23] E[(Xε(t))2]∼2Dt c(t/tc)α ast→ ∞.(41) The convergence in (35) implies thatX ε in (40) becomes the RLfBm in (15) asεvanishes, Xε →X RLfBm asε→0 +.(42) C. FPTs IfX ε is the bounded speed process in eit...

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    Appendix C: Numerical simulation details We now describe the numerical methods used in sec- tion IV

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    sBm Figures 2, 3, and 4 use the exact FPT distribution formula for the unbounded speed sBm in (26) and the following exact FPT distribution formula for the bounded speed sBm, P(τε > t) =S ε(tc(t/tc)α), whereS ε(t) is the following FPT distribution for the bounded speed sBm (and bounded speed RLfBm) when α= 1 [20, 33], Sε(t) = 1−Θ(t−t min) e−κ +κ t/tminZ 1...

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