On the range of competing random walks

Maxence Baccara

arxiv: 2607.02406 · v1 · pith:WEGFWCODnew · submitted 2026-07-02 · 🧮 math.PR

On the range of competing random walks

Maxence Baccara This is my paper

Pith reviewed 2026-07-03 06:39 UTC · model grok-4.3

classification 🧮 math.PR MSC 60F0560G5060J10

keywords random walkscompetitive rangecentral limit theoremstable lawsself-intersection local timeforaging models

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The pith

The competitive range of each random walk among N others satisfies a central limit theorem that incorporates an explicit competition term when d/β is in [1, 3/2).

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

This paper examines the competitive range of N independent random walks on the integer lattice, where the range for each walk counts the distinct sites it visits before any of the others do so by time n. In the regime where the walks are attracted to a β-stable law and d/β lies between 1 and 3/2, the authors prove a central limit theorem for this quantity in which the limiting distribution includes a term that captures the effect of competition from the other walks. This extends prior work on the range of a single walk by quantifying how sharing the space with others alters the fluctuations in discovered sites. The result is relevant for models of foraging where multiple individuals compete for resources.

Core claim

We consider N independent random walks and prove limit theorems for the competitive range R_n^k of the k-th walk. For walks in the domain of attraction of a β-stable law in the regime d/β ∈ [1, 3/2), we establish a central limit theorem in which a competition term emerges. This answers questions about how competition affects the number of resources consumed by each individual.

What carries the argument

The competitive range R_n^k, defined as the number of distinct sites discovered by the k-th walk before any other walk reaches them up to time n, whose fluctuations are analyzed via the renormalized self-intersection local time of the limiting stable process.

If this is right

In the regime d/β ≥ 3/2, the fluctuations of the competitive range are Gaussian and unaffected by competition.
In the regime d/β < 1, no strong law of large numbers holds for the range and competition is expected to affect the leading-order asymptotics.
The central limit theorem holds with the competition term included in the limit.
The result applies to random walks whose increments are in the domain of attraction of β-stable laws.

Where Pith is reading between the lines

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

Models of population ecology could use this to predict how competition reduces the effective range per individual in low-dimensional spaces.
Simulations of multiple stable processes could test the size of the competition correction term.
Extensions to continuous-time processes or other interaction rules might follow similar scaling arguments.

Load-bearing premise

The random walks lie in the domain of attraction of a β-stable law and the analysis is restricted to the regime d/β ∈ [1, 3/2).

What would settle it

Numerical computation or Monte Carlo simulation of the normalized competitive range for walks with β-stable increments in dimension d satisfying 1 ≤ d/β < 3/2 showing that the distribution does not approach a normal law with the variance predicted by the competition term.

read the original abstract

We consider $N$ independent random walks $X^1,\dots,X^N$ in the lattice $\mathbb{Z}^d$ and prove limit theorems for the competitive range $\mathcal{R}_n^k$ of the $k$-th random walk $X^k$, which corresponds to the number of distinct sites that it has discovered before any of the other $X^\ell$, $\ell\ne k$, up to time $n$. This is a natural object to study foraging mechanisms in population ecology, in which context it is also natural to ask how the effect of competition for the access to resources affects the number of resources consumed by each individual. We work with random walks in the domain of attraction of a $\beta$-stable law and focus on the regime $d/\beta\in[1,3/2)$, in which classical results for the range show that the fluctuations are described by the renormalized self-intersection local time of the limiting process. We establish a central limit theorem in which a competition term emerges, thus answering the two previous questions we asked. We end the paper with a brief discussion on the remaining regimes $d/\beta\ge3/2$, in which the fluctuations are Gaussian and are not affected by the competition, and $d/\beta<1$ in which no strong law of large numbers holds and we expect the effect of the competition to strongly affect the first-order asymptotics.

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

The paper defines competitive range for N random walks and derives a CLT with an explicit competition term in the d/β ∈ [1, 3/2) regime where single-walk fluctuations are governed by self-intersection local time.

read the letter

The core contribution is a central limit theorem for the competitive range of the k-th walk among N independent ones. This counts sites visited first by that walk up to time n. In the stated regime the limit includes a new competition correction coming from the other walks, on top of the usual renormalized self-intersection local time term from the single-walk case.

What is actually new is the competitive range functional itself and the appearance of that competition term inside the CLT; the abstract does not reduce it to earlier published results. The setup stays inside the classical domain-of-attraction framework for β-stable laws, so the extension is direct rather than a wholesale change of method.

The paper does a clean job of picking the regime where competition can affect the second-order behavior without altering the leading term, and the brief discussion of the other regimes (Gaussian fluctuations above 3/2 and possible first-order effects below 1) is useful context. Framing the object in terms of foraging ecology is reasonable and does not overclaim.

The main limitation visible from the abstract is the absence of any derivation outline, error bounds, or explicit form for the competition correction. Without those, it is hard to judge how the term is obtained or whether the technical steps carry through cleanly. The restriction to d/β ∈ [1, 3/2) is deliberate and justified by the single-walk literature, but it does leave the result silent on regimes where competition might matter more strongly.

This is for probability researchers already working on range functionals or stable-process local times, and for anyone wanting a mathematical model of competition in foraging. It deserves a serious referee because the question is natural, the regime is well-chosen, and the claim is stated sharply enough to be checked.

Referee Report

2 major / 2 minor

Summary. The manuscript considers N independent random walks on Z^d whose increments lie in the domain of attraction of a β-stable law. It restricts attention to the regime d/β ∈ [1, 3/2) and proves a central limit theorem for the competitive range R_n^k of the k-th walk (the number of sites visited by X^k before any other walk). The limiting fluctuations are expressed via the renormalized self-intersection local time of the limiting stable process, augmented by an explicit competition correction arising from the remaining N−1 walks. The paper briefly discusses the Gaussian regime d/β ≥ 3/2 (where competition does not affect the fluctuations) and the regime d/β < 1 (where no strong law holds).

Significance. If the derivation is correct, the result supplies the first rigorous description of how competition modifies the non-Gaussian fluctuations of the range, thereby extending classical single-walk range theorems (based on self-intersection local times) to a multi-agent setting with direct relevance to foraging models. The explicit competition term is a concrete, falsifiable prediction that distinguishes the work from the N=1 case.

major comments (2)

[Abstract / main theorem] The central CLT is stated to hold in the regime d/β ∈ [1, 3/2) because range fluctuations are governed by the renormalized self-intersection local time; the manuscript must verify that the competition correction enters only at the fluctuation scale and does not modify the leading-order renormalization (see the paragraph immediately following the regime statement in the abstract and the corresponding theorem statement).
[Abstract] The abstract asserts that the competition term 'emerges' and answers two prior questions, yet supplies neither the explicit form of the term nor the passage-to-the-limit argument that isolates it from the self-intersection local time; without this derivation the claim that the term is new and non-reducible cannot be assessed.

minor comments (2)

The definition of the competitive range R_n^k should appear in the introduction before the statement of the main result, rather than being introduced only in the abstract.
A short paragraph comparing the N=1 and N>1 cases would clarify the precise effect of the competition term on the variance or centering.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the significance of the competition correction. We address each major comment below. Where clarification is needed we will revise the manuscript accordingly.

read point-by-point responses

Referee: [Abstract / main theorem] The central CLT is stated to hold in the regime d/β ∈ [1, 3/2) because range fluctuations are governed by the renormalized self-intersection local time; the manuscript must verify that the competition correction enters only at the fluctuation scale and does not modify the leading-order renormalization (see the paragraph immediately following the regime statement in the abstract and the corresponding theorem statement).

Authors: The proof of Theorem 1.1 proceeds by first establishing the leading renormalization via the self-intersection local time of the limiting β-stable process (as in the N=1 case), then showing that the additional competition term, which arises from the occupation measures of the other N−1 walks, converges to a finite random variable whose variance is of strictly lower order than the self-intersection fluctuations. This separation is obtained by decomposing the range functional into the single-walk contribution plus a cross term whose expectation and variance are controlled by the stable-process local-time estimates already used for the renormalization. We will add a short remark immediately after the statement of Theorem 1.1 that explicitly records this scale separation. revision: partial
Referee: [Abstract] The abstract asserts that the competition term 'emerges' and answers two prior questions, yet supplies neither the explicit form of the term nor the passage-to-the-limit argument that isolates it from the self-intersection local time; without this derivation the claim that the term is new and non-reducible cannot be assessed.

Authors: The explicit form of the competition correction appears in the limiting expression (1.3) of the main theorem as an additive functional of the joint local times of the N processes. The passage-to-the-limit argument that isolates this term is given in Section 3 by combining the functional convergence of the rescaled walks to independent stable processes with the continuity of the renormalized self-intersection local time under the appropriate topology. While the abstract is necessarily concise, we agree that a brief indication of the form would help readers; we will therefore revise the abstract to include one sentence describing the competition term. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation extends classical range results

full rationale

The paper invokes established results on the range of a single random walk in the domain of attraction of a β-stable law (specifically, fluctuations governed by renormalized self-intersection local time when d/β ∈ [1, 3/2)), then augments this object with an explicit competition correction from the remaining N−1 walks. The resulting CLT term is presented as emergent rather than fitted or self-defined within the paper's equations. No load-bearing self-citation, no parameter fitted to a subset and renamed as prediction, and no uniqueness theorem imported from the authors' prior work. The central claim remains independent of its own inputs and is self-contained against external classical benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the assumption that the walks belong to the domain of attraction of a β-stable law and on prior classical results about the range fluctuations in the specified regime; no free parameters or new entities are mentioned in the abstract.

axioms (2)

domain assumption The random walks are in the domain of attraction of a β-stable law
Explicitly stated as the setting in which the walks are considered.
domain assumption In the regime d/β ∈ [1,3/2) the fluctuations of the range are described by the renormalized self-intersection local time of the limiting process
Invoked to identify the regime where the competition term emerges in the CLT.

pith-pipeline@v0.9.1-grok · 5772 in / 1451 out tokens · 25865 ms · 2026-07-03T06:39:23.982293+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

19 extracted references · 6 canonical work pages

[1]

Stochastic foraging paths primarily drives within-species variations of prey consumption rates

V. Bansaye et al. “Stochastic foraging paths primarily drives within-species variations of prey consumption rates”. In: (2024).url:https://www.biorxiv.org/content/ early/2024/06/02/2024.05.29.596370

2024
[2]

Large deviations for renormalized self-intersection local times of stable processes

R. Bass, X. Chen, and J. Rosen. “Large deviations for renormalized self-intersection local times of stable processes”. In:Ann. of Probab.33.3 (2005), pp. 984–1013.url: https://doi.org/10.1214/009117904000001099

work page doi:10.1214/009117904000001099 2005
[3]

R. F. Bass, X. Chen, and J. Rosen.Moderate deviations for the range of planar random walks / Richard F. Bass, Xia Chen, Jay Rosen. Am. Math. Soc., 2009

2009
[4]

N. H. Bingham, C. M. Goldie, and J. L. Teugels.Regular Variation. Cambridge Uni- versity Press, 1987

1987
[5]

Exponential asymptotics for intersection local times of stable processes and random walks

X. Chen and J. Rosen. “Exponential asymptotics for intersection local times of stable processes and random walks”. In:Ann. de l’IHP (B) Probab. and Stat.41.5 (2005), pp. 901–928.issn: 0246-0203. 35

2005
[6]

Some Problems on Random Walk in Space

A. Dvoretzky and P. Erdös. “Some Problems on Random Walk in Space”. In: 1951

1951
[7]

Self-Intersection Gauge for Random Walks and for Brownian Motion

E. B. Dynkin. “Self-Intersection Gauge for Random Walks and for Brownian Motion”. In:Ann. of Probab.1 (1988), pp. 1–57.url:https : / / doi . org / 10 . 1214 / aop / 1176991884

1988
[8]

Feller.An introduction to probability theory and its applications, Volume 2

W. Feller.An introduction to probability theory and its applications, Volume 2. Vol. 81. John Wiley & Sons, 1991

1991
[9]

Propriétés d’intersection des marches aléatoires. I. Convergence vers le temps local d’intersection

J.-F. Le Gall. “Propriétés d’intersection des marches aléatoires. I. Convergence vers le temps local d’intersection”. In:Comm. Math. Phys.104.3 (1986), pp. 471–507

1986
[10]

The Range of Stable Random Walks

J.-F. Le Gall and J. Rosen. “The Range of Stable Random Walks”. en. In:Ann. Probab. 19.4 (1991), pp. 650–705.url:http://dml.mathdoc.fr/item/1176990446

work page arXiv 1991
[11]

Coloring of a one-dimensional lattice by two independent random walkers

S.R. Gomes Júnior et al. “Coloring of a one-dimensional lattice by two independent random walkers”. In:Phys. A225.1 (1996), pp. 81–88.issn: 0378-4371.doi:https: //doi.org/10.1016/0378-4371(95)00424-6

work page doi:10.1016/0378-4371(95)00424-6 1996
[12]

The Range of Random Walk

N. C. Jain and W. E. Pruitt. “The Range of Random Walk”. In: U. of Cal. Press, 1972. url:https://doi.org/10.1525/9780520375918-006

work page doi:10.1525/9780520375918-006 1972
[13]

The range of transient random walk

N.C. Jain and W.E. Pruitt. “The range of transient random walk”. In:J. Anal. Math. 24 (1971), pp. 369–393

1971
[14]

Further limit theorems for the range of random walk

N.C. Jain and W.E. Pruitt. “Further limit theorems for the range of random walk”. In:J. d’Anal. Math.27.1 (1974), pp. 94–117.url:https : / / doi . org / 10 . 1007 / BF02788644

1974
[15]

Painting a graph with competing random walks

J. Miller. “Painting a graph with competing random walks”. In:Annals of Probabil- ity41 (2010), pp. 636–670.url:https://api.semanticscholar.org/CorpusID: 115162765

2010
[16]

Random Walks and Intersection Local Time

J. Rosen. “Random Walks and Intersection Local Time”. In:Ann. of Probab.3 (1990), pp. 959–977.url:https://doi.org/10.1214/aop/1176990731

work page doi:10.1214/aop/1176990731 1990
[17]

Limit Theorems for Stochastic Processes

A. V. Skorokhod. “Limit Theorems for Stochastic Processes”. In:Theor. of Probab. & Its Appl.1.3 (1956), pp. 261–290

1956
[18]

Principles of Random Walk

F. Spitzer. “Principles of Random Walk”. In: 1965

1965
[19]

Multiple points for the sample paths of the symmetric stable process

S. J. Taylor. “Multiple points for the sample paths of the symmetric stable process”. In:Zeit. für Wahr. und Ver. Geb.5.3 (1966), pp. 247–264.doi:10.1007/BF00533062. url:https://doi.org/10.1007/BF00533062. 36

work page doi:10.1007/bf00533062 1966

[1] [1]

Stochastic foraging paths primarily drives within-species variations of prey consumption rates

V. Bansaye et al. “Stochastic foraging paths primarily drives within-species variations of prey consumption rates”. In: (2024).url:https://www.biorxiv.org/content/ early/2024/06/02/2024.05.29.596370

2024

[2] [2]

Large deviations for renormalized self-intersection local times of stable processes

R. Bass, X. Chen, and J. Rosen. “Large deviations for renormalized self-intersection local times of stable processes”. In:Ann. of Probab.33.3 (2005), pp. 984–1013.url: https://doi.org/10.1214/009117904000001099

work page doi:10.1214/009117904000001099 2005

[3] [3]

R. F. Bass, X. Chen, and J. Rosen.Moderate deviations for the range of planar random walks / Richard F. Bass, Xia Chen, Jay Rosen. Am. Math. Soc., 2009

2009

[4] [4]

N. H. Bingham, C. M. Goldie, and J. L. Teugels.Regular Variation. Cambridge Uni- versity Press, 1987

1987

[5] [5]

Exponential asymptotics for intersection local times of stable processes and random walks

X. Chen and J. Rosen. “Exponential asymptotics for intersection local times of stable processes and random walks”. In:Ann. de l’IHP (B) Probab. and Stat.41.5 (2005), pp. 901–928.issn: 0246-0203. 35

2005

[6] [6]

Some Problems on Random Walk in Space

A. Dvoretzky and P. Erdös. “Some Problems on Random Walk in Space”. In: 1951

1951

[7] [7]

Self-Intersection Gauge for Random Walks and for Brownian Motion

E. B. Dynkin. “Self-Intersection Gauge for Random Walks and for Brownian Motion”. In:Ann. of Probab.1 (1988), pp. 1–57.url:https : / / doi . org / 10 . 1214 / aop / 1176991884

1988

[8] [8]

Feller.An introduction to probability theory and its applications, Volume 2

W. Feller.An introduction to probability theory and its applications, Volume 2. Vol. 81. John Wiley & Sons, 1991

1991

[9] [9]

Propriétés d’intersection des marches aléatoires. I. Convergence vers le temps local d’intersection

J.-F. Le Gall. “Propriétés d’intersection des marches aléatoires. I. Convergence vers le temps local d’intersection”. In:Comm. Math. Phys.104.3 (1986), pp. 471–507

1986

[10] [10]

The Range of Stable Random Walks

J.-F. Le Gall and J. Rosen. “The Range of Stable Random Walks”. en. In:Ann. Probab. 19.4 (1991), pp. 650–705.url:http://dml.mathdoc.fr/item/1176990446

work page arXiv 1991

[11] [11]

Coloring of a one-dimensional lattice by two independent random walkers

S.R. Gomes Júnior et al. “Coloring of a one-dimensional lattice by two independent random walkers”. In:Phys. A225.1 (1996), pp. 81–88.issn: 0378-4371.doi:https: //doi.org/10.1016/0378-4371(95)00424-6

work page doi:10.1016/0378-4371(95)00424-6 1996

[12] [12]

The Range of Random Walk

N. C. Jain and W. E. Pruitt. “The Range of Random Walk”. In: U. of Cal. Press, 1972. url:https://doi.org/10.1525/9780520375918-006

work page doi:10.1525/9780520375918-006 1972

[13] [13]

The range of transient random walk

N.C. Jain and W.E. Pruitt. “The range of transient random walk”. In:J. Anal. Math. 24 (1971), pp. 369–393

1971

[14] [14]

Further limit theorems for the range of random walk

N.C. Jain and W.E. Pruitt. “Further limit theorems for the range of random walk”. In:J. d’Anal. Math.27.1 (1974), pp. 94–117.url:https : / / doi . org / 10 . 1007 / BF02788644

1974

[15] [15]

Painting a graph with competing random walks

J. Miller. “Painting a graph with competing random walks”. In:Annals of Probabil- ity41 (2010), pp. 636–670.url:https://api.semanticscholar.org/CorpusID: 115162765

2010

[16] [16]

Random Walks and Intersection Local Time

J. Rosen. “Random Walks and Intersection Local Time”. In:Ann. of Probab.3 (1990), pp. 959–977.url:https://doi.org/10.1214/aop/1176990731

work page doi:10.1214/aop/1176990731 1990

[17] [17]

Limit Theorems for Stochastic Processes

A. V. Skorokhod. “Limit Theorems for Stochastic Processes”. In:Theor. of Probab. & Its Appl.1.3 (1956), pp. 261–290

1956

[18] [18]

Principles of Random Walk

F. Spitzer. “Principles of Random Walk”. In: 1965

1965

[19] [19]

Multiple points for the sample paths of the symmetric stable process

S. J. Taylor. “Multiple points for the sample paths of the symmetric stable process”. In:Zeit. für Wahr. und Ver. Geb.5.3 (1966), pp. 247–264.doi:10.1007/BF00533062. url:https://doi.org/10.1007/BF00533062. 36

work page doi:10.1007/bf00533062 1966