On the range of competing random walks

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.