Coupling Brownian loop soups and random walk loop soups at all polynomial scales
Wei Qian
TL;DR
This work removes the prior restriction that the Lawler–Trujillo Ferreras coupling between Brownian loop soups and random walk loop soups only applied to loops above a mesoscopic threshold, establishing a coupling at all polynomial scales for all dimensions $d\ge 1$. The authors construct a refined coupling using a new time-slicing sequence $\{a_n\}$ satisfying $a_n=\frac{2n}{d}+O(1)$ and leverage a multidimensional KMT coupling to align long and short loop segments across the Brownian and random-walk settings, for both continuous- and discrete-time variants. The result yields a one-to-one correspondence between loops with time-length above a scale determined by $\theta\in(0,2)$, with tight proximity bounds $|t_\gamma-t_{\widetilde{\gamma}}|\le c N^{-2}$ and $\sup_{0\le s\le 1}|\gamma(s t_\gamma)-\widetilde{\gamma}(s t_{\widetilde{\gamma}})|\le c N^{-1}\log N$, extending the coupling to mesoscopic loops and to all dimensions. This advances the applicability of loop-soup couplings to scaling-limit analyses and chaos approximations, enabling precise, global control of loop structures at all polynomial scales. The results also fill gaps for $d\ge 3$ where convergence of random-walk loop soups to Brownian loop soups had been less explicit, with potential implications for related conformal or geometric limit theorems.
Abstract
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 $N^{θ-2}$ for $θ\in(2/3,2)$. In particular, there is no control on mesoscopic loops with time length less than $N^{-4/3}$ (i.e.\ roughly diameter less than $N^{-2/3}$). In this paper, we find a simple way to remove the restriction $θ>2/3$, 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 $\mathbb{Z}^d$ and Brownian loop soups on $\mathbb{R}^d$).