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

Lawler and Trujillo Ferreras [13] constructed a well-known coupling between the Brownian loop soups on R2 and the (discrete-time) random walk loop soups on Z2 (one rescales the random walk loops by 1∕N, their time parametrizations by 1∕(2N2), and lets N→∞), which led to numerous applications. It nevertheless only holds for loops with time length at least Nθ−2 for θ∈(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 in [21] to Zd with d≥3, for loops with time length at least Nθ−2, for θ∈(2d∕(d+4),2). In this paper, we find a simple way to remove the restriction θ>2d∕(d+4), so that such a coupling works for all θ∈(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 Zd, for d≥1. As an intermediate step, we also establish a KMT coupling between the continuous-time random walk bridge on Zd and the Brownian bridge on Rd.