Coupling Brownian loop soups and random walk loop soups at all polynomial scales

Abstract

Lawler and Trujillo Ferreras constructed a well-known coupling between the Brownian loop soups on ${\mathbb{R}}^2$ and the (discrete-time) random walk loop soups on ${\mathbb{Z}}^2$ (one rescales the random walk loops by $1/N$, their time parametrizations by $1/(2N^2)$, and lets $N\to \infty$), which led to numerous applications. It nevertheless only holds for loops with time length at least $N^{\hat{I}¸-2}$ for $\hat{I}¸\in (2/3,2)$. In particular, there is no control on mesoscopic loops with time length less than $N^{-4/3}$ (i.e. roughly diameter less than $N^{-2/3}$). This coupling was subsequently extended by Sapozhnikov and Shiraishi to ${\mathbb{Z}}^d$ with $d\ge 3$, for loops with time length at least $N^{\hat{I}¸-2}$, for $\hat{I}¸\in (2d/(d+4),2)$. In this paper, we find a simple way to remove the restriction $\hat{I}¸>2d/(d+4)$, so that such a coupling works for all $\hat{I}¸\in (0,2)$, i.e. for loops at all polynomial scales. We establish couplings for both discrete-time and continuous-time random walk loop soups on ${\mathbb{Z}}^d$, for $d\ge 1$. As an intermediate step, we also establish a KMT coupling between the continuous-time random walk bridge on ${\mathbb{Z}}^d$ and the Brownian bridge on ${\mathbb{R}}^d$.

19 pages