Coupling Brownian loop soups and random walk loop soups at all polynomial scales
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| Format: | Preprint |
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2026
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| _version_ | 1866911383701946368 |
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| author | Qian, Wei |
| author_facet | Qian, Wei |
| contents | 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 $N^{-2/3}$). This coupling was subsequently extended by Sapozhnikov and Shiraishi to $\mathbb{Z}^d$ with $d\ge 3$, for loops with time length at least $N^{θ-2}$, for $θ\in(2d/(d+4),2)$.
In this paper, we find a simple way to remove the restriction $θ>2d/(d+4)$, so that such a coupling works for all $θ\in (0,2)$, 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 $d\ge 1$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_02992 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Coupling Brownian loop soups and random walk loop soups at all polynomial scales Qian, Wei Probability 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 $N^{-2/3}$). This coupling was subsequently extended by Sapozhnikov and Shiraishi to $\mathbb{Z}^d$ with $d\ge 3$, for loops with time length at least $N^{θ-2}$, for $θ\in(2d/(d+4),2)$. In this paper, we find a simple way to remove the restriction $θ>2d/(d+4)$, so that such a coupling works for all $θ\in (0,2)$, 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 $d\ge 1$. |
| title | Coupling Brownian loop soups and random walk loop soups at all polynomial scales |
| topic | Probability |
| url | https://arxiv.org/abs/2601.02992 |