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 in $\mathbb{R}<sup>2$ and the random walk loop soups on $\mathbb{Z}<sup>2$ (one rescales the random walk loops by $1/N$, their time parametrizations by $1/(2N^2)$, and let $N\to \infty$), which led to numerous applications. It nevertheless only holds for loops with time length at least $N<sup>{θ-2}$ for $θ\in(2/3,2)$. In particular, there is no control on mesoscopic loops with time length less than $N<sup>{-4/3}$ (i.e.\ roughly diameter less than $N<sup>{-2/3}$). In this paper, we find a simple way to remove the restriction $θ&gt;2/3$, so that such a coupling works for all $θ\in (0,2)$, i.e. for loops at all polynomial scales. We also establish this coupling in any dimension $d\ge 1$ (i.e. for random walk loop soups on $\mathbb{Z}<sup>d$ and Brownian loop soups on $\mathbb{R}<sup>d$).