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 $1/N$, 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 $N^{\theta-2}$ for $\theta \in(2/3,2)$. 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 $N^{\theta-2}$, for $\theta \in(2d/(d+4),2)$. In this paper, we find a simple way to remove the restriction $\theta>2d/(d+4)$, 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 $d\ge 1$.