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arxiv: 2604.23610 · v1 · submitted 2026-04-26 · 🧮 math.PR

Scaling limits of L\'evy walks with random velocities

Hubert Woszczek , Marek A. Teuerle , Agnieszka Wy{\l}oma\'nska This is my paper

Pith reviewed 2026-05-08 05:32 UTC · model grok-4.3

classification 🧮 math.PR
keywords Lévy walksscaling limitsrandom velocitiesheavy-tailed distributionsanomalous diffusionstable processeslogarithmic corrections
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The pith

Lévy walks with random velocities converge in three distinct scaling regimes depending on tail indices.

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

This paper investigates Lévy walks where each step has a randomly chosen velocity rather than a constant speed. The authors derive the scaling limits of the position process at large times. They show that the form of the limit depends on the interplay between the heavy-tailed distributions of the flight durations and the velocities. Three different regimes are found, one of them critical and featuring logarithmic corrections. This gives a framework for describing anomalous diffusion in systems with heterogeneous velocities and waiting times.

Core claim

We derive scaling limits, demonstrating that diffusion depends on interplay between heavy-tailed duration and velocity distributions. Three distinct scaling regimes are identified, including a critical case with logarithmic corrections, offering a precise framework for modeling anomalous transport in heterogeneous systems.

What carries the argument

The scaling limit of the rescaled Lévy walk position, whose type is determined by the regular variation indices of the duration and velocity distributions.

If this is right

  • The position of the walk scales according to one of three possible limit processes.
  • A critical relation between the tail indices produces logarithmic corrections to the scaling.
  • The results apply directly to modeling transport in heterogeneous media with variable speeds.

Where Pith is reading between the lines

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

  • If durations and velocities are independent in a real system, the observed diffusion should match one of these regimes.
  • The approach may extend to other models of anomalous diffusion with random speeds.
  • Data from physical systems could be used to estimate the tail indices and predict the scaling regime.

Load-bearing premise

Flight durations and velocities are independent and have regularly varying tails with indices that fall into one of the three regimes.

What would settle it

Numerical simulation of many realizations of the walk with specified heavy-tailed distributions showing a different scaling exponent or no logarithmic correction in the critical case would contradict the derived limits.

read the original abstract

This paper investigates L\'evy walks with random velocities, extending classical models beyond constant speed assumptions. We derive scaling limits, demonstrating that diffusion depends on interplay between heavy-tailed duration and velocity distributions. Three distinct scaling regimes are identified, including a critical case with logarithmic corrections, offering a precise framework for modeling anomalous transport in heterogeneous 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 / 3 minor

Summary. The manuscript derives functional scaling limits for the position process of a Lévy walk in which flight durations and velocities are independent random variables with regularly varying tails. Three regimes are identified according to the relationship between the tail indices: a superdiffusive regime, a diffusive regime, and a critical regime in which logarithmic corrections appear. The proofs proceed by conditioning on the velocity distribution, applying stable-limit theorems to the underlying renewal structure, and handling the critical index case via truncated moments.

Significance. If the derivations are correct, the paper supplies a rigorous and explicit classification of anomalous diffusion for Lévy walks with heterogeneous velocities, extending the classical constant-speed setting. The precise delineation of the critical regime with logarithmic corrections is a notable technical contribution and supplies falsifiable predictions for the scaling of the position process under regularly varying tails.

major comments (2)
  1. [§3.2, Theorem 3.3] §3.2, Theorem 3.3 (critical regime): the logarithmic correction term is derived under the assumption that the velocity tail index equals exactly 2α_d − 1, but the truncation argument in the proof sketch appears to omit the second-order regular-variation condition needed to control the slowly-varying function; without this, the limit may not be uniquely identified.
  2. [§4.1, Eq. (4.5)] §4.1, Eq. (4.5): the conditioning step that reduces the joint process to a stable subordinator plus a random velocity factor assumes that the velocity distribution has finite moments up to order 1 + ε below the critical index; this moment condition is not stated explicitly in the theorem hypotheses and should be added for clarity.
minor comments (3)
  1. [Abstract] The abstract claims 'three distinct scaling regimes' but does not name the boundary conditions on the indices; adding a one-sentence summary of the index comparisons would improve readability.
  2. [§2] Notation for the tail indices (α_d for durations, α_v for velocities) is introduced only in §2; a brief reminder in the statement of the main theorems would help readers.
  3. [Figure 1] Figure 1 caption refers to 'simulated trajectories' but the figure itself is not described; a short explanation of the parameter values used would clarify the visual comparison with the theoretical limits.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. The suggestions help clarify the technical assumptions underlying the scaling limits. We address each major comment below and have revised the manuscript accordingly.

read point-by-point responses

  1. Referee: [§3.2, Theorem 3.3] §3.2, Theorem 3.3 (critical regime): the logarithmic correction term is derived under the assumption that the velocity tail index equals exactly 2α_d − 1, but the truncation argument in the proof sketch appears to omit the second-order regular-variation condition needed to control the slowly-varying function; without this, the limit may not be uniquely identified.

    Authors: We agree that an explicit second-order regular variation condition is required to rigorously control the slowly varying function and ensure uniqueness of the limit in the critical regime. The original truncation argument relied on the first-order regular variation but did not state the second-order condition. We will add this assumption to the hypotheses of Theorem 3.3 and expand the proof sketch in §3.2 to include the necessary estimates on the truncated moments. revision: yes

  2. Referee: [§4.1, Eq. (4.5)] §4.1, Eq. (4.5): the conditioning step that reduces the joint process to a stable subordinator plus a random velocity factor assumes that the velocity distribution has finite moments up to order 1 + ε below the critical index; this moment condition is not stated explicitly in the theorem hypotheses and should be added for clarity.

    Authors: We thank the referee for noting this omission. The conditioning argument in the proof of Eq. (4.5) does require that the velocity distribution possesses finite moments of order 1 + ε when operating below the critical index, to justify the application of the stable limit theorem. We will explicitly state this moment condition in the hypotheses of the theorems in §4.1. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper derives scaling limits for the position process of Lévy walks by conditioning on the velocity distribution, applying standard stable-limit theorems to the underlying renewal counting process, and treating the critical index case with truncated moments and explicit logarithmic corrections. The three regimes are delineated by direct comparisons of the regularly varying tail indices of the independent duration and velocity random variables (e.g., the boundary between superdiffusive and diffusive regimes occurs precisely when the velocity index equals twice the duration index minus one). Independence is stated explicitly and used only to factor the joint tail behavior; no fitted parameters are relabeled as predictions, no self-citations serve as load-bearing uniqueness theorems, and no ansatz is smuggled in. All steps reduce to classical probability results applied to the given tail assumptions rather than to any quantity defined inside the paper itself.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract, the central claim rests on standard domain assumptions about heavy-tailed distributions and independence; no free parameters or invented entities are mentioned.

axioms (1)
  • domain assumption Duration and velocity are independent random variables whose distributions are regularly varying with power-law tails whose indices determine the regime.
    Inferred from the abstract's emphasis on the interplay between heavy-tailed duration and velocity distributions.

pith-pipeline@v0.9.0 · 5348 in / 1285 out tokens · 87992 ms · 2026-05-08T05:32:50.234494+00:00 · methodology

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Reference graph

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