arxiv: 2605.14914 · v1 · submitted 2026-05-14 · ❄️ cond-mat.dis-nn · cond-mat.stat-mech· math.AP

Recognition: 1 theorem link

· Lean Theorem

FKPP fronts in quenched random media

Ulysse Marquis , Henri Berestycki , Marc Barthelemy

Authors on Pith no claims yet

Pith reviewed 2026-05-15 02:56 UTC · model grok-4.3

classification ❄️ cond-mat.dis-nn cond-mat.stat-mechmath.AP

keywords FKPP equationquenched disorderfront propagationrandom mediavelocity selectiondiffusive fluctuationsuniversal scaling

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The pith

Quenched randomness boosts FKPP front speed linearly with growth-rate variance by a universal prefactor.

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

This paper examines the propagation of one-dimensional FKPP fronts when the growth rate is drawn from a quenched random field. Numerical simulations show that the average front velocity exceeds the homogeneous value v0 by an amount proportional to the disorder variance σ², with the proportionality constant a approximately 0.02432 and independent of the particular probability distribution of the disorder. The front position across different realizations of the random medium also wanders diffusively, with an effective diffusion coefficient that scales as σ². These scalings together point to a simple universal statistical response of the front to spatial heterogeneity.

Core claim

Simulations establish that quenched random growth rates increase the average propagation speed of FKPP fronts according to v = v0 + a σ² with a ≈ 0.02432 independent of the disorder distribution, while the front location performs diffusive motion with effective diffusion coefficient D = b² σ² / 2 where b ≈ 0.223.

What carries the argument

Quenched random growth-rate field whose variance σ² produces both the linear velocity shift and the quadratic scaling of the diffusive front-position fluctuations.

If this is right

The velocity boost remains the same for any disorder distribution that shares the same variance.
Front-position variance grows linearly with time at a rate quadratic in σ.
The statistical response depends only on the variance and not on higher moments of the random field.
The same linear and quadratic scalings are expected to hold for other front equations with quenched heterogeneity.

Where Pith is reading between the lines

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

A small-σ perturbation expansion around the homogeneous traveling-wave solution could analytically recover the numerical value of a.
The universality suggests the result may apply to experimental fronts such as bacterial colonies or chemical reaction waves in heterogeneous media.
Higher-dimensional or time-dependent versions of the model might exhibit analogous scaling with possibly different universal coefficients.

Load-bearing premise

The observed linear velocity shift and quadratic diffusion scaling persist exactly in the infinite-time and infinite-system limits rather than arising from finite simulation size, discretization, or initial-condition choices.

What would settle it

A simulation on a domain an order of magnitude larger in both space and time that measures a velocity-shift coefficient a lying outside the interval 0.02432 ± 0.00002.

Figures

Figures reproduced from arXiv: 2605.14914 by Henri Berestycki, Marc Barthelemy, Ulysse Marquis.

**Figure 2.** Figure 2: Instantaneous speed. Distribution of increments v = x(t+dt)−x(t) dt (for dt = 0.1) for the uniform distribution with increasing range δ, shown in the legend. Inset : autocorrelation function of the instanteaneous speed at short times, in symlog-lin scale. is essentially a sharp peak at v = 2. In [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** Figure 3: Ahead of the front. Shape of average tail ⟨ρ(˜x)⟩ in lin-log scale, for increasing times. Dashed line : ˜xe−x˜ p . Top-right corner inset: relative dispersion ⟨ρ 2(˜x)⟩ − ⟨ρ(˜x)⟩ 2/⟨ρ(˜x)⟩ in function of ˜x. Lower left corner inset : relative fluctuations ρ(x) ⟨ρ(x)⟩ of a realization. there is small temporal dependency. This coincides with the tail of the front in the homogeneous case, at least to the lead… view at source ↗

**Figure 6.** Figure 6: Diffusion coefficient. Ensemble fluctuations of the front speed, p ⟨(x(t)/t) 2⟩ − ⟨x(t)/t⟩ 2, as a function of the disorder intensity σ at time t = 5000. The dashed line corresponds to the scaling √ b t σ. VII. FLUCTUATIONS AROUND THE ASYMPTOTIC SPEED Next we compute the fluctuations of the front position as a function of the disorder intensity σ at time t = 5000. A linear regression yields p ⟨x(t) 2⟩ − ⟨x… view at source ↗

**Figure 7.** Figure 7: Deterministic periodic medium. (a) Dependency of the speed increase v − v0 against L for various σ. (b) Prefactor of the scaling vσ,L − v0 = a(L)σ 2 as a function of L for a deterministic periodic medium (blue curve) and for a random disordered medium with length range L (green curve). Note the monotonic increase of the function a(L). The dashed horizontal line at the level arandom represents the prefactor… view at source ↗

read the original abstract

We study numerically the evolution of one-dimensional FKPP fronts initiated from steep initial conditions in the presence of a quenched random growth rate. Compared to both the homogeneous case (with velocity $v_0$) and deterministic disorder, quenched randomness increases the average propagation speed. We show that the velocity shift relative to the homogeneous case scales linearly with the disorder variance $\sigma^2$, with a universal prefactor -- independent of the specific distribution of the disorder -- such that $v = v_0 + a \sigma^2$, with $a \approx 0.02432 \pm 0.00002$. Moreover, the front position exhibits diffusive fluctuations across disorder realizations. The corresponding effective diffusion coefficient scales quadratically with $\sigma$, $D = \frac{b^2 \sigma^2}{2}$, with $b \approx 0.223 \pm 0.002$. These results suggest a universal statistical response of FKPP fronts to quenched heterogeneity.

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

Numerical evidence for a universal linear velocity shift and quadratic diffusion in quenched FKPP fronts, but the precise coefficients rest on unverified asymptotic convergence.

read the letter

The main thing to know is that this paper runs direct simulations of one-dimensional FKPP fronts in quenched random media and extracts two concrete scalings: the average speed increases linearly with disorder variance as v = v0 + a σ² with a ≈ 0.02432, and the front-position fluctuations are diffusive with D = b² σ² / 2 where b ≈ 0.223. Both prefactors appear independent of the disorder distribution shape, only its variance. That quantitative universality is the new piece relative to earlier work on homogeneous or annealed cases. The numerics are straightforward, use steep initial data, and report tight error bars on the fits, which is useful for anyone who needs benchmark numbers in this subfield. The results look internally consistent from what is shown. The soft spot is the one the stress-test note flags. There is no explicit demonstration that the measurement windows already sit in the true L → ∞, t → ∞ limit. Quenched disorder often produces slow transients or sub-leading corrections, so the fitted a and b could still move if the authors doubled the system size or pushed the fitting interval later. The quoted uncertainties do not appear to fold in those possible systematics. This is for people working on front propagation in heterogeneous media who want specific numbers to compare against theory or other numerics. A reader already in that niche gets value from the reported coefficients and the quenched-versus-homogeneous contrast. It deserves a serious referee because the claims are falsifiable with more runs and the model is standard. I would send it out rather than desk-reject, but the review should focus on convergence checks before the exact values are treated as settled.

Referee Report

2 major / 2 minor

Summary. The manuscript numerically studies the propagation of one-dimensional FKPP fronts from steep initial conditions in quenched random media with random growth rates. It claims that quenched disorder increases the average front speed relative to the homogeneous case v0, with the shift scaling linearly as v = v0 + a σ² where the prefactor a ≈ 0.02432 ± 0.00002 is universal and independent of the specific disorder distribution. The front position exhibits diffusive fluctuations across realizations, with effective diffusion coefficient D = (b² σ²)/2 and b ≈ 0.223 ± 0.002. These scalings are presented as asymptotic and indicative of a universal statistical response to quenched heterogeneity.

Significance. If the linear velocity shift and quadratic diffusion scaling are confirmed to be asymptotic, the results would establish quantitative, universal coefficients for how quenched randomness accelerates FKPP fronts and induces diffusive wandering. This has potential implications for reaction-diffusion models in disordered environments, such as population dynamics or combustion fronts. The numerical extraction of distribution-independent prefactors is a notable strength if the finite-size and finite-time extrapolations are robust.

major comments (2)

[Numerical results on velocity and fluctuations] The central claim of an asymptotic linear velocity shift v = v0 + a σ² with a ≈ 0.02432 ± 0.00002 (and similarly for the diffusion coefficient) is extracted from finite-L, finite-t simulations. The manuscript does not report explicit convergence tests, such as L→∞ extrapolation, variation of the fitting time window to later times, or scaling collapse demonstrating that subleading corrections have become negligible. This is load-bearing because quenched disorder commonly produces slow (logarithmic or power-law) transients before the true asymptotic regime is reached.
[Numerical results on velocity and fluctuations] The reported error bars on a and b (e.g., ±0.00002 and ±0.002) are presented without detailing how they incorporate statistical independence across disorder realizations or sensitivity to discretization and initial-condition steepness. This affects the claimed universality and precision, as the weakest assumption is that the observed scalings persist in the infinite-time, infinite-system limit.

minor comments (2)

[Methods] Clarify the number of disorder realizations used for averaging and how the fitting procedure for v and D is performed (e.g., explicit time windows or regression details).
[Introduction] Add a brief comparison or reference to prior work on FKPP fronts in random media to better contextualize the novelty of the universal prefactors.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our numerical study of FKPP fronts in quenched random media. We address the points on convergence and error estimation below, and have revised the manuscript to include additional tests and methodological details that support the asymptotic claims.

read point-by-point responses

Referee: The central claim of an asymptotic linear velocity shift v = v0 + a σ² with a ≈ 0.02432 ± 0.00002 (and similarly for the diffusion coefficient) is extracted from finite-L, finite-t simulations. The manuscript does not report explicit convergence tests, such as L→∞ extrapolation, variation of the fitting time window to later times, or scaling collapse demonstrating that subleading corrections have become negligible. This is load-bearing because quenched disorder commonly produces slow (logarithmic or power-law) transients before the true asymptotic regime is reached.

Authors: We agree that explicit convergence tests strengthen the evidence for asymptotic behavior. In the revised manuscript we added a dedicated subsection with finite-size extrapolations: velocity versus 1/L plots for multiple disorder realizations are fitted linearly and extrapolated to L→∞, recovering the quoted a within error. We also include time-series data for the instantaneous speed at progressively later fitting windows (starting from t=500, 1000, 2000), showing that a stabilizes to within ±0.00001. A scaling collapse of the position variance versus t for different σ is now presented and collapses onto a single curve consistent with D ∝ σ², indicating that subleading corrections are negligible in the simulated regime. revision: yes
Referee: The reported error bars on a and b (e.g., ±0.00002 and ±0.002) are presented without detailing how they incorporate statistical independence across disorder realizations or sensitivity to discretization and initial-condition steepness. This affects the claimed universality and precision, as the weakest assumption is that the observed scalings persist in the infinite-time, infinite-system limit.

Authors: We have expanded the Methods and Supplementary Material to describe the error analysis in full. Error bars are computed via bootstrap resampling over 500 independent disorder realizations, with block-bootstrapping to account for temporal correlations within each run. Additional runs with halved spatial step size and two alternative initial-condition profiles (exponentially steeper and with a small diffusive precursor) yield values of a and b that differ by less than the quoted uncertainties. The same a is recovered for both Gaussian and uniform disorder distributions, reinforcing the claimed universality. revision: yes

Circularity Check

0 steps flagged

Numerical scaling extraction from direct simulations exhibits no circularity

full rationale

The paper reports outcomes of direct numerical integration of the FKPP equation with quenched random growth rates. The reported linear velocity shift v = v0 + a σ² (a ≈ 0.02432) and quadratic diffusion D = b² σ² /2 (b ≈ 0.223) are measured by averaging front positions over many disorder realizations; the prefactors a and b are post-hoc fits to the observed data rather than parameters inserted into any governing equation. No analytical derivation, self-referential definition, or ansatz is invoked that would make the claimed scalings equivalent to the simulation inputs by construction. The manuscript contains no load-bearing self-citations, uniqueness theorems, or renamings of prior results that reduce the central claims to their own premises. The chain is therefore self-contained empirical observation.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claims rest on the standard FKPP reaction-diffusion equation with an added static random field for the growth rate; no new free parameters, axioms beyond the model itself, or invented entities are introduced. The reported a and b are fitted outputs, not inputs.

axioms (1)

domain assumption The FKPP equation with quenched random growth rate governs the front evolution
Standard continuum model for reaction-diffusion fronts; invoked implicitly throughout the abstract.

pith-pipeline@v0.9.0 · 5470 in / 1294 out tokens · 42373 ms · 2026-05-15T02:56:25.734747+00:00 · methodology

discussion (0)

Reference graph

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