On the range of competing random walks

Abstract: We consider $N$ independent random walks $X<sup>1,\dots,X<sup>N$ in the lattice $\mathbb{Z}<sup>d$ and prove limit theorems for the competitive range $\mathcal{R}_n<sup>k$ of the $k$-th random walk $X<sup>k$, which corresponds to the number of distinct sites that it has discovered before any of the other $X<sup>\ell$, $\ell\ne k$, 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 $β$-stable law and focus on the regime $d/β\in[1,3/2)$, 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 $d/β\ge3/2$, in which the fluctuations are Gaussian and are not affected by the competition, and $d/β&lt;1$ in which no strong law of large numbers holds and we expect the effect of the competition to strongly affect the first-order asymptotics.

• The paper establishes that the mean behavior (LLN) of the competitive range matches that of the classical range, indicating no leading order effect from competition.
• It shows that competition significantly alters fluctuations, producing non-Gaussian limits due to additive intersection local time contributions.
• The analysis uses dyadic decomposition and inclusion-exclusion strategies to rigorously control intersection effects, informing models in ecology and multi-agent systems.

Summary: On the Range of Competing Random Walks (2607.02406)

Problem Overview and Motivations

This paper investigates the competitive range of independent random walks on $\mathbb{Z}^d$: for $N$ i.i.d. random walks $X^1,\dots,X^N$, the competitive range of the $k$-th walk up to time $n$, denoted $_n^k$, is the number of distinct sites first discovered by $X^k$ before any others among the $N$ walks. This object is motivated primarily by mathematical ecology, modeling competitive foraging among multiple agents on static resources.

The classical range, $|X^k(0,n)|$, which counts all unique sites visited by $X^k$ up to time $N$0, has an extensive literature, with complete law of large numbers (LLN) and central limit theorem (CLT) results for various types of random walks and different spatial dimension/transience regimes. By contrast, the competitive range introduces nontrivial correlations through competition, fundamentally altering the fluctuation behavior in certain regimes.

Main Results

Assumptions and Regimes

The analysis focuses on i.i.d. random walks in the domain of attraction of a strictly $N$1-stable Lévy process on $N$2, with scale function $N$3, and restricts attention to the parameter regime $N$4, which is characterized by non-Gaussian, highly nontrivial fluctuation phenomena for the classical range.

Law of Large Numbers for Competitive Range

The primary LLN result is that, for each $N$5,

$N$6

in $N$7 (and almost surely if the slowly varying part $N$8 for all $N$9), where $X^1,\dots,X^N$0 is the truncated Green's function of the walk. This is identical to the LLN for the classical range: competition has no effect at the leading order. The proof employs an inclusion-exclusion decomposition, expressing $X^1,\dots,X^N$1 in terms of its own range and "ordered intersection" terms involving subsets of other walks, with these intersection terms shown to be negligible in the LLN limit.

Central Limit Theorem and Fluctuations

The strong qualitative difference between range and competitive range emerges at the CLT level. For $X^1,\dots,X^N$2:

$X^1,\dots,X^N$3

where $X^1,\dots,X^N$4 is the renormalized self-intersection local time of the limiting stable process $X^1,\dots,X^N$5, and $X^1,\dots,X^N$6 is the intersection local time between two independent limit processes. Competition manifests only in the fluctuations, through additive terms in the limit distribution involving these intersection local times. Notably, the fluctuations are not Gaussian; instead, their limit is a nontrivial, highly non-Gaussian law constructed from intersection local times.

Extension to Other Regimes

• For $X^1,\dots,X^N$7, the fluctuations become Gaussian and the effect of competition vanishes at the fluctuation scale: competitive and total range CLTs coincide, both exhibiting classic diffusive scaling.
• For $X^1,\dots,X^N$8, the range ceases to satisfy an LLN—random walks are so recurrent that strong concentration fails. The author conjectures, supported by arguments and exact proofs for simple random walks in $X^1,\dots,X^N$9, that in this regime competition does appear already at the leading order: the limiting law has a support modified by the earliest hitting profiles among all walks.

Technical Approach

A key ingredient is a dyadic decomposition of the competitive range, generalizing Le Gall's classical decomposition for the range of a random walk. The proof uses inclusion-exclusion for first-visit times, combined with sharp asymptotic control over intersection local times for both discrete random walks and their stable process scaling limits. The technique of mollified intersection local times, together with uniform integrability arguments, enables handling of both the self-intersection and pairwise intersection contributions at the fluctuation scale.

The limitations on $k$0 arise essentially from the parameter region where intersection local time exponents lead to non-Gaussian, nontrivially correlated fluctuations. The cutoff and approximation arguments require careful application of Skorokhod/Donsker invariance principles, Fourier analysis on the transition kernels, and properties of regularly varying functions.

Theoretical and Practical Implications

From a theoretical perspective, this work demonstrates the subtleties introduced by competition in an otherwise well-understood probabilistic system. While the mean behavior is largely unchanged (competition appears only at the fluctuation scale, and not at all in some asymptotic regimes), the limiting fluctuations become fundamentally more complex and cannot be captured by simple Gaussian approximations in the nontrivial regime.

Practically, this suggests that in models of competitive exploration or resource consumption where agents are sufficiently "spaced out" (high $k$1), competitive effects average out, but in intermediate regimes ($k$2) the exploitation "fairness" is both random and highly sensitive to the complex overlap structure of the agents' trajectories. For theoretical ecology or multi-agent search, these results provide scaling laws and structural predictions for how competitive effects emerge as the environmental or stochastic parameters change.

Future Directions

• Extension to non-identically distributed, dependent, or non-Markovian walks: The methodology extends but would require technical refinements (the paper indicates possibilities for non-i.i.d. walks).
• Heavy-tailed or Lévy flights with $k$3: In this regime, competition modifies leading asymptotics, suggesting distinct phenomena for highly recurrent (e.g., fractional Brownian or infinite-variance) foragers.
• Applications to spatial games or invasion processes: The structure of first-passage competition could inform models in spatial game theory or multi-type branching random walks.
• Large deviation and moderate deviation principles for the competitive range: Could competitive corrections induce nonstandard rate functions or phenomena not present for single-walker range?

Conclusion

This paper rigorously delineates the scaling regimes in which competition between random walkers meaningfully alters the statistics of visited sites. In the intermediate regime $k$4, competition fundamentally reshapes the fluctuation law of the number of first-discovered sites, introducing non-Gaussian, intersection-driven limiting variables, while leaving the mean unchanged. Outside this regime, competition either does not affect leading or fluctuation-scale asymptotics or dominates already at the leading order in highly recurrent cases. The work bridges probabilistic, analytical, and ecological perspectives, and its techniques and conclusions are broadly relevant for interacting particle systems on lattices.