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

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