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Spontaneous stochasticity and the Armstrong-Vicol passive scalar

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Pith's one-line read This paper proves spontaneous stochasticity is a measure-selection phenomenon, and that the Armstrong–Vicol passive scalar exhibits both Lagrangian and Eulerian forms of it.

desk verdict A genuinely useful framework for Eulerian spontaneous stochasticity, but Theorem 1 has a real proof gap; the Armstrong-Vicol application rests on solid external work and is not the main risk. read the letter →

arxiv 2504.15795 v1 pith:T4DMUXY5 submitted 2025-04-22 physics.flu-dyn

W. Ruffenach , E. Simonnet , N. Valade This is my paper · ORCID
classification physics.flu-dyn
keywords spontaneousstochasticitypassivescalaranomalousdissipationselectionprinciplemeasurefractalhomogenisationBebutovflowrenormaliseddiffusivity
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

This paper tries to establish two things. First, in a singular inviscid limit, the absence of a selection principle is not a pathology: it is the phenomenon of spontaneous stochasticity, defined as a continuous observable taking two distinct limit values along different vanishing-regularisation subsequences. Second, the Armstrong–Vicol passive scalar is spontaneously stochastic in both the Lagrangian and the Eulerian sense. The measure-theoretic core is a proof that every probability measure on the attainable set of the inviscid system is the inviscid-limit statistic of some admissible regularisation, so spontaneous stochasticity becomes a measure-selection problem whenever the inviscid system is non-unique. A sympathetic reader would care because it turns 'which solution is selected?' into 'which probability measure over solutions is selected?', with universality classes classified by the ergodic properties of regularisations.

What carries the argument

The machinery has two levels. In the general theory, the central object is the regularisation curve $\gamma(\tau)=\Phi_t[f(\cdot,1/\tau)]x_0$ living in the space $\mathrm{Reg}$, acted on by the Bebutov flow; Birkhoff averages along this flow yield the set $\mathcal{M}(\gamma)$ of subsequential statistics, and the paper proves $\mathcal{M}=\mathcal{M}_0=\mathcal{P}(S_0)$, with $\mathcal{M}_0$ the probability measures on the attainable set. This turns universality classes into genericity sets $\mathcal{B}_\mu$ of regularisations sharing the same limiting measure. For the Armstrong–Vicol model, the load-bearing object is the renormalised diffusivity sequence $\kappa_{m-1}=\kappa_m+c_0\varepsilon_m^{2\beta}/\kappa_m$ with hypergeometric scale separation $\varepsilon_m^{-1}=\lceil\Lambda^{q^m/(q-1)}\rceil$, together with the homogenisation constraints that make the alternating shear-flow cascade produce anomalous diffusion; Lemma 2 ties the dissipation observable to the renormalised diffusivity at scale $m_\star$, and Lemma 3 bounds $\kappa_+/\kappa_-$ from below, forcing the ratio $A_+/A_-$ above 1.

What would settle it

Check well-posedness of the interpolated field $f_\varepsilon=\theta_\varepsilon F_\varepsilon+(1-\theta_\varepsilon)f_0$ for a continuous $f_0$ with nonunique inviscid solutions: if $\dot x=f_\varepsilon(x)$ has two solutions for arbitrarily small $\varepsilon$, Theorem 1's construction is not a valid regularisation. For Theorem 3, compute the renormalised diffusivity sequence with $\varepsilon_0^{-1}=\lceil\Lambda^{1/(q-1)}\rceil$ and evaluate $A_+/A_-$ from the dissipation observable; if the lower bound (80) is violated so that $A_+/A_-\le 1$ for all admissible observables, the claimed Eulerian spontaneous stochasticity does not follow.

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Extended reading notes

Core claim

The central claim is that spontaneous stochasticity — defined through the regularisation function $A(\kappa)=O(\Phi_t[f(\cdot,\kappa)]x_0)$ as $-\infty<\liminf_{\kappa\to 0}A(\kappa)<\limsup_{\kappa\to 0}A(\kappa)<+\infty$ — is exactly equivalent to finite-time trajectory splitting and to the lack of a selection principle, and that this equivalence is measure-theoretically complete. Theorem 1 states that the space $\mathcal{M}$ of all probability measures obtainable from regularisations coincides with $\mathcal{M}_0=\mathcal{P}(S_0)$, the set of all probability measures supported on the attainable set of the inviscid system; hence any statistical behaviour compatible with the inviscid system is realisable by some well-posed regularisation. For the Armstrong–Vicol model, Theorem 3 proves that with $m_\star=0$ and $\Lambda$ sufficiently large the dissipation observable satisfies $A_+/A_- \ge e^{-4\varepsilon} A_+^{m_\star}/A_-^{m_\star} > 1$, so the passive scalar exhibits Eulerian spontaneous stochasticity, in addition to the Lagrangian spontaneous stochasticity that follows from anomalous diffusion.

Load-bearing premise

The central claim stands on two premises: that the interpolated regularisations built in Theorem 1 are well-posed even though only continuity of $f_0$ is assumed, and that the Armstrong–Vicol estimates proven for $m_\star\ge 1$ extend to the $m_\star=0$ convention used in Theorem 3; if either fails, the corresponding proof collapses.

Editorial extensions

If this is right

  • Absence of a selection principle is equivalent to spontaneous stochasticity and to finite-time trajectory splitting, so unbounded finite-time Lyapunov exponents in the inviscid limit are a direct signature of the phenomenon.
  • Because $\mathcal{M}=\mathcal{M}_0$, any probability measure on the set of attainable inviscid solutions is realisable as a limiting statistic; universality classes are exactly the genericity sets $\mathcal{B}_\mu$ and are distinguished by the ergodic properties of regularisations.
  • As soon as the inviscid system is non-unique, there exist regularisations for which the system is spontaneously stochastic; a necessary condition is that trajectories hit a non-Lipschitz singular set in finite time, detectable through Dini-type directional derivatives.
  • The Armstrong–Vicol passive scalar is Eulerian spontaneously stochastic: the dissipation rate has two distinct limit values as diffusivity vanishes, and it is also Lagrangian spontaneously stochastic through anomalous diffusion.
  • For the hypergeometric renormalised diffusivity sequence, a well-defined probability measure only emerges on the log-log scale; the limiting distribution of the renormalised diffusivity is non-Dirac and encodes the measure-selection information.

Reading between the lines

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

  • If the measure-selection theorem extends to infinite-dimensional settings, then numerical and experimental searches for a selection principle in turbulence should target measure-valued fixed points of renormalisation rather than a single trajectory; shell models with isolated Hölder singularities are a minimal, computationally accessible testbed.
  • For the Armstrong–Vicol model, Eulerian spontaneous stochasticity predicts that under-resolved simulations should show the dissipation plateau jumping between two values as resolution or diffusivity is tuned; computing $A(\kappa)$ along the renormalised sequence and measuring $A_+/A_-$ would test inequality (40) directly.
  • The conjecture that every non-renormalisable regularisation becomes renormalisable after a continuous reparametrisation suggests that the apparent ambiguity of the inviscid limit may be a gauge-like redundancy; if true, the physical statistics are the invariants under such reparametrisations, which parallels renormalisation-group fixed-point universality.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. The paper proposes a general framework for spontaneous stochasticity (SpSt) in finite-dimensional dynamical systems, defines SpSt through the liminf/limsup of a regularisation function, proves an equivalence with the lack of a selection principle, and develops a measure-theoretic description of the inviscid limit based on Bebutov flows and ergodic limits. Its central structural claim is Theorem 1: if M0 is the space of probability measures on the attainable set of the inviscid system, then M=M0, i.e. every probability measure on S0 can be attained as the inviscid-limit statistic of some well-posed regularisation. The paper also gives a necessary condition for nonuniqueness via Dini-type derivatives, applies the framework to the Armstrong-Vicol passive scalar, and claims in Theorem 3 that this model exhibits Eulerian spontaneous stochasticity in the sense of Definition 1. Part I contains a historical review, a description of the Armstrong-Vicol construction, and numerical experiments on anomalous diffusion and Lagrangian spontaneous stochasticity.

Significance. If the central claims hold, the paper would provide a rigorous reformulation of spontaneous stochasticity as a measure-selection phenomenon and would establish a clean dichotomy: non-uniqueness of the inviscid system is sufficient, through the M=M0 theorem, for the existence of regularisations exhibiting SpSt. The application to the Armstrong-Vicol model is also significant, since it identifies the absence of a selection principle with Eulerian spontaneous stochasticity. The paper is honest in crediting the Armstrong-Vicol result, and it provides detailed appendices, a reproducible numerical algorithm for the multiscale flow, and a large amount of historical context. However, the two pillars of the theoretical part — Theorem 1 and the paper's own proof of Theorem 3 — contain load-bearing gaps that need repair; the conclusions may be true, but they are not established as written.

major comments (4)
  1. [Section 5.3.2 / Appendix 11.6] The construction of the regularised vector field f_epsilon = theta_epsilon F_epsilon + (1 - theta_epsilon) f0 is not shown to satisfy Hypothesis (H0). Under (H0), f0 is only continuous, so the interpolation is generally only continuous; Peano's theorem gives existence but not uniqueness, and the assertion that g_epsilon is 'the unique solution' of dx/dt = f_epsilon(x) is unsupported. This matters because in the intended applications f0 is non-Lipschitz (otherwise S0 would be a singleton). The same problem affects the convex-combination step f_theta = a f_x + (1-a) f_y used to obtain arbitrary mixtures of Dirac measures: well-posedness is not an inherited property under convex combinations of well-posed vector fields. The proof needs either a modified construction that provably yields a unique solution for every kappa>0, or an additional hypothesis and an Osgood/Lipschitz estimate for f0.
  2. [Section 5.3.2 / Theorem 1 and Corollary 1] The statement M0 = co(E) is false as written because co(E) is defined as the set of finite convex combinations, while P(S0) is generally larger. The intended statement is that M0 is the closed convex hull of E. The proof only places finite convex combinations of Dirac masses in M and then says 'taking the closure'. But M is an infinite union of the compact sets M(gamma), and no closedness of M is established. Density of co(E) in M0 does not imply M = M0 unless M is closed. Corollary 1 has a related defect: the universal regularisation gamma_un is built from two-point combinations theta delta_x + (1-theta) delta_y only, and such measures are not weak-* dense in P(S0) when S0 has more than two points. A countable dense family of finitely supported measures and a diagonal argument would be required to obtain M(gamma_un) = M0.
  3. [Section 7 / Appendix 11.13] The proof of Theorem 3 depends crucially on the convention m* = 0 and on the lower bound (80), which is used to force A+/A- > 1. The Armstrong-Vicol estimates invoked in Lemma 2 are proved in [3,4] for m* >= 1. Appendix 11.13 simply asserts that 'all necessary estimates in [3] also hold' for the convention epsilon_0^{-1} = ceil(Lambda^{1/(q-1)}), without proof. This is a load-bearing assertion: Lemma 2, Lemma 3, and inequality (80) all rely on it. In addition, the statement of Theorem 3 does not explicitly include the analyticity condition (51) that is used in the cited estimates. The authors should either provide the missing verification for the m* = 0 extension, or present Theorem 3 explicitly as a corollary of Proposition 5.5 of [4] rather than as an independent proof.
  4. [Sections 5.1-5.6 and Section 7] Definition 1 and the framework of Sections 5.3-5.5 are formulated for finite-dimensional phase space H = R^d, with compactness, Krein-Milman, and Prokhorov arguments used for probability measures on a compact subset of R^d. Section 5.6 explicitly states that the PDE extension is only formal and that the authors 'will not pursue this further'. Nevertheless, Theorem 3 states that the passive scalar equation (1) is spontaneously stochastic 'in the sense of Definition 1'. To make this a theorem, one needs an infinite-dimensional version of Definition 1 and of Hypothesis (H0), including a phase space on which the chosen observable is continuous. Without such a formulation, the claim that (1) exhibits SpSt is a corollary of the Armstrong-Vicol result [4], not a theorem of the framework developed in this paper.
minor comments (4)
  1. [Section 3.1] The reference 'Fig. .1' should read 'Fig. 1'.
  2. [Equation (14)] The norm in the final display is missing a closing parenthesis: it should be ||nabla theta^kappa||^2_{L^2((0,t) x T^2)}.
  3. [Section 4.2] The text describing Figure 4 refers to 'dotted black lines' and 'solid black line' in a way that is hard to reconcile with the figure caption; please harmonise the notation and the captions.
  4. [Section 7] Using epsilon both for the small parameter in (39) and for the scale parameter epsilon_m is confusing; consider renaming the small parameter to eta to avoid collision.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Theorem 1 is built from independent abstract theorems, and Theorem 3 is an explicit corollary of external Armstrong–Vicol estimates; the identified weaknesses are proof gaps, not circular reductions.

full rationale

The paper's derivation chain does not reduce to its own inputs. Theorem 1 (M = M0, Section 5.3.2 and Appendix 11.6) is proved from independent mathematical anchors: Peano/Arzela-Ascoli compactness, Stone-Weierstrass approximation, Krein-Milman, Prokhorov, and explicit constructions of regularisations. The regularisation space Reg and the attainable-measure space M are defined in terms of Hypothesis (H0), but the proof that Dirac measures and finite convex combinations lie in M is a genuine construction, not an equivalence smuggled in through the definitions. Theorem 3 (Section 7) is explicitly presented as a corollary of the Armstrong–Vicol estimates [3,4], which are external works by different authors; the ratio bound A+/A- >= e^{-4 epsilon} A+_m*/A-_m* > 1 is derived from those estimates plus a new, sharper diffusivity-sequence lemma, not fitted from the target quantity. The paper also explicitly acknowledges that its Definition 1 is equivalent to the absence of a selection principle (Section 5.2, Appendix 11.1, and the conclusion), so applying the name 'Eulerian spontaneous stochasticity' to the Armstrong–Vicol non-uniqueness result is a terminological identification, not a circular renaming. The numerical comparison of the Richardson scaling in Section 4.2 uses an arbitrary multiplicative constant and does not enter the proofs; it is illustrative support only. The self-citations in the reference list ([19], [82]) appear in historical or contextual remarks and are not load-bearing. The genuine weaknesses in the manuscript are mathematical proof gaps rather than circularity: in Appendix 11.6 the interpolated fields f_epsilon = theta_epsilon F_epsilon + (1 - theta_epsilon) f0 are asserted to be well-posed 'by construction', although under (H0) only continuity of f0 is assumed; the proof of Theorem 1 concludes M = M0 by taking a closure without first establishing that the union M is closed (though Corollary 1 supplies a fixed-gamma argument that would repair this); and Appendix 11.13 asserts without proof that the Armstrong–Vicol estimates extend to the m* = 0 convention. Each of these is an omitted or incomplete verification, not a reduction of a claimed prediction to a fitted input or to a self-citation chain. Accordingly, no circular step meets the evidentiary standard of quoting a specific equation that makes the output equal to the input by construction.

Assumptions & free parameters 6 free parameters · 8 assumptions · 2 invented entities

The paper's central results rest on the Armstrong-Vicol construction and estimates, taken as external input from [3, 4]; Theorem 3 is a corollary of those results. The framework parts use standard tools (Peano, Osgood, Kneser, Krein-Milman, Prokhorov) plus the explicit assumption (H0) on regularisations. The nonstandard, paper-specific inputs are: the assertion that Armstrong-Vicol estimates extend to m* = 0 (Appendix 11.13), the closeness argument between the idealised renormalised sequence and the true sequence (Appendix 11.12), and the conjectured existence of a nontrivial pullback attractor for the renormalised sequence (Section 8). Numerically, the hand-chosen parameters (Lambda = 2.5, beta = 1.2, M = 3) lie outside the Armstrong-Vicol hypotheses. No new physical entities are introduced; the non-renormalisable regularisation and level-1 SpSt are formal constructs without independent empirical handles.

free parameters (6)
  • Numerical scale-separation parameter Lambda = Lambda = 2.5 (Table 1)
    Chosen by hand for computational feasibility; below the Lambda >= 27 threshold required by the Armstrong-Vicol theorem, so the numerics probe a regime the theorem does not cover.
  • Numerical regularity parameter beta = beta = 1.2 (Table 1)
    Picked inside (1, 4/3); with Lambda = 2.5 the Armstrong-Vicol hypotheses are not strictly satisfied, as the paper states.
  • Number of resolved scales M = M = 3
    Limited by the several-hundred-GB storage of 2048^2 fields; three to four scales is the maximum achievable resolution.
  • Effective large-time diffusivity in Figure 4 (right) = about 8e-3
    Fitted line used to identify the effective diffusive regime; the authors admit a convincing measurement is not available.
  • Richardson-superdiffusion prefactor in Figure 4 (right) = arbitrary multiplicative constant
    The comparison line matches the predicted exponent only up to a freely chosen prefactor, weakening the fit as evidence.
  • Permissible-set parameter A in Theorem 3 = free within (1, A_max] per constraint (39)
    Proof parameter controlling the renormalised-diffusivity ratio kappa+/kappa- >= c1 A^2; introduced ad hoc to force A+/A- > 1, not fitted to data.
assumptions (8)
  • domain assumption Armstrong-Vicol fractal-homogenisation construction and estimates: Theorem 1.1, estimate (5.72), Lemmas 3.3 and 3.4 of [3, 4]
    Enter at equations (1)-(7), Section 7, and Appendix 11.10; the paper's Theorem 3 and the numerical interpretation inherit these results without re-derivation.
  • domain assumption Hypothesis (H0): (P_kappa) well-posed for kappa > 0 and f(.,kappa) converges to f0 uniformly
    Used in Definition 1, Property 1 (Appendix 11.1), and in the definition of the regularisation space Reg (equation (21)).
  • ad hoc to paper Armstrong-Vicol estimates extend to the m* = 0 convention epsilon_0^{-1} = ceil(Lambda^{1/(q-1)})
    Asserted in Appendix 11.13 without proof; necessary for the lower bound (80) and hence for Theorem 3 as proven in this paper.
  • standard math Energy identity (2) bounds A(kappa) = kappa ||nabla theta_kappa||^2 between 0 and ||theta_0||^2 / 2
    Used to ensure the SpSt observable satisfies -infinity < A- < A+ < +infinity for the Armstrong-Vicol model.
  • standard math Peano existence theorem, Osgood uniqueness criterion, and Dini-derivative characterisation of Lipschitz continuity
    Basis of Theorem 2 (Appendix 11.8) and of the nonuniqueness discussion in Section 5.5.
  • standard math Kneser's theorem (attainable set S0 compact and connected), Krein-Milman, and Prokhorov's theorem
    Used in Property 2, Property 3, and Theorem 1 (Appendices 11.4 and 11.6).
  • domain assumption Drivas-Eyink Lagrangian fluctuation-dissipation relation: anomalous diffusion is equivalent to Lagrangian spontaneous stochasticity
    Used in Sections 4.2 and 5.6 to equate the numerically measured dissipation plateau with Lagrangian SpSt; cited from [31], assumed valid for the Armstrong-Vicol field.
  • domain assumption Richardson-type superdiffusive exponent sigma^2 proportional to s^{1+(1+alpha+gamma)/(1-alpha)} from [4]
    Quoted in Section 4.2 as 'expected' and 'not proven there'; used for the comparison line in Figure 4 (right).
invented entities (2)
  • Non-renormalisable regularisation (gamma with M(gamma) not a singleton)
    purpose: Classifies regularisations whose inviscid-limit statistics depend on the subsequence; motivates the ambient-measure choice and the conjecture in Section 5.3.3
    A formal mathematical construct (Section 5.3.2, Corollary 1); no falsifiable prediction outside the framework.
  • Level-1 spontaneous stochasticity (nontrivial Bebutov pushforward at measure level)
    purpose: Hypothetical stronger form of SpSt beyond Definition 1
    Explicitly conjectural in Section 5.3.3; no independent handle is provided.

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Pith. "Pith review of Spontaneous stochasticity and the Armstrong-Vicol passive scalar." pith.science (2026). https://pith.science/paper/T4DMUXY5

@misc{pith2026250415795,
  author       = {Pith},
  title        = {Pith review of: Spontaneous stochasticity and the Armstrong-Vicol passive scalar},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/T4DMUXY5}},
  note         = {Machine review of arXiv:2504.15795}
}
read the original abstract

Spontaneous stochasticity refers to the emergence of intrinsic randomness in deterministic systems under singular limits, a phenomenon conjectured to be fundamental in turbulence. Armstrong and Vicol \citep{AV23,AV24} recently constructed a deterministic, divergence-free multiscale vector field arbitrarily close to a weak Euler solution, proving that a passive scalar transported by this field exhibits anomalous dissipation and lacks a selection principle in the vanishing diffusivity limit. {\it This work aims to explain why this passive scalar exhibits both Lagrangian and Eulerian spontaneous stochasticity.} Part I provides a historical overview of spontaneous stochasticity, details the Armstrong-Vicol passive scalar model, and presents numerical evidence of anomalous diffusion, along with a refined description of the Lagrangian flow map. In Part II, we develop a theoretical framework for Eulerian spontaneous stochasticity. We define it mathematically, linking it to ill-posedness and finite-time trajectory splitting, and explore its measure-theoretic properties and the connection to RG formalism. This leads us to a well-defined {\it measure selection principle} in the inviscid limit. This approach allows us to rigorously classify universality classes based on the ergodic properties of regularisations. To complement our analysis, we provide simple yet insightful numerical examples. Finally, we show that the absence of a selection principle in the Armstrong-Vicol model corresponds to Eulerian spontaneous stochasticity of the passive scalar. We also numerically compute the probability density of the effective renormalised diffusivity in the inviscid limit. We argue that the lack of a selection principle should be understood as a measure selection principle over weak solutions of the inviscid system.

Figures

Figures reproduced from arXiv: 2504.15795 by the authors.

Figure 1
Figure 1. A sketch of homogenisation at scale ϵm. The time ∆t is the time needed to see the shear at scale ϵm, namely to have a displacement along y of order ϵm giving ϵm ∼ √ 2κ∆t. It gives the effective diffusivity along x: κm,x = ⟨L 2 x ⟩ ∆t = κ + a 2 mϵ 2 m∆t = κ + a 2 mϵ 4 m κ . Then one rotates the field after waiting some time τm ≫ ∆t, that is waiting long enough for homogenisation to occur. Then one proceeds again in t… view at source ↗
Figure 2
Figure 2. Left: Norm of the vector field at a given time and zoom on a given region to highlight the multiscale nature of the flow. Parameters used in the simulations [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Schematic representation of the non-local in time dependence of [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figures from the paper (10 more)
Figure 4
Figure 4. Figure 4: (Left) Diffusion rate of the passive scalar field over time for different values of κ. Since the initial condition is independent of κ, the initial dissipation rate is proportional to κ. However, after a transient regime of order 102 τM ≃ 2τ1, the dissipation rate beco…
Figure 5
Figure 5. Figure 5: A sketch view of trajectory splitting in the singular limit [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]
Figure 6
Figure 6. Figure 6: A tale of spontaneous stochasticity. We present our results graph [PITH_FULL_IMAGE:figures/full_fig_p015_6.png]
Figure 7
Figure 7. Figure 7: We focus on deterministic regularisations, represented by the red [PITH_FULL_IMAGE:figures/full_fig_p017_7.png]
Figure 8
Figure 8. Figure 8: A(κ) = x κ (1) solution of (34) for κ uniformally distributed in [10−4 , 10−1 ]: in red A1 and in black A2. Both measures concentrate on ±a with a ≈ 0.544. two regularisations A1 and A2, the pushforward measures are not equal but their support is the same: the first pe…
Figure 9
Figure 9. Figure 9: Solutions selected in the limit κ → 0 for the system ˙x = √ |x|, x(0) = −c 2 for two different regularisations, in red by additive noise: the system is not SpSt (A+ = A− ), in blue using the regularisation (35) with T ∈ [1, 3 2 ] ∪ [2, 5 2 ] ∪ [3, 7 2 ] uniformally dis…
Figure 10
Figure 10. Figure 10: Distribution of the solutions of (36) at [PITH_FULL_IMAGE:figures/full_fig_p020_10.png]
Figure 12
Figure 12. Figure 12: Upper panel: standard deviation of log10 κ0 as a function of β and Λ. Regime I corresponds to κ0 distributions having a simply-connected compact support, for 1.14 ⪅ β ⪅ 1.21, and regime II to a bimodal qualitative behaviour of the κ0 distribution. Three probability de…
Figure 13
Figure 13. Figure 13: The diffusivity sequence κm in loglog scale for four different values of β using Λ = 128. One can notice that as β → 4 3 , the slope tends to zero while the intercept diverges to minus infinity [PITH_FULL_IMAGE:figures/full_fig_p035_13.png]
Figure 14
Figure 14. Figure 14: Values of κ0 as a function of κM = κ (in log10 scale) for β = 1.10 < β ⋆ (Λ = 128). The renormalised diffusivity sequence becomes wild, with un￾bounded behaviour unless κM ∈ K is imposed. 35 [PITH_FULL_IMAGE:figures/full_fig_p035_14.png]

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