Three-dimensional Brownian loop soup clusters
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866911404491014144 |
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| author | Jego, Antoine Lupu, Titus |
| author_facet | Jego, Antoine Lupu, Titus |
| contents | We study Brownian loop soup clusters in $\mathbb{R}^3$ for an arbitrary intensity $α>0$. We show the existence of a phase transition for the presence of unbounded clusters and study its basic properties. In particular, we show that, when $α$ is sufficiently large, almost surely all the loops are connected into a single cluster. Such a phenomenon is not observed in discrete percolation-type models. In addition, we prove the existence of a one-arm exponent and compare the clusters with the finite-range system obtained by imposing lower and upper bounds on the diameter of the loops.
Finally, we provide a toolbox concerning the Brownian loop measure in $\mathbb{R}^d$, $d \ge 3$. In particular, we derive decomposition formulas by rerooting the loops in specific ways and show that the loop measure is conformally invariant, generalising results of [Lup18] in dimension 1 and [LW04] in dimension 2. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_04840 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Three-dimensional Brownian loop soup clusters Jego, Antoine Lupu, Titus Probability Mathematical Physics We study Brownian loop soup clusters in $\mathbb{R}^3$ for an arbitrary intensity $α>0$. We show the existence of a phase transition for the presence of unbounded clusters and study its basic properties. In particular, we show that, when $α$ is sufficiently large, almost surely all the loops are connected into a single cluster. Such a phenomenon is not observed in discrete percolation-type models. In addition, we prove the existence of a one-arm exponent and compare the clusters with the finite-range system obtained by imposing lower and upper bounds on the diameter of the loops. Finally, we provide a toolbox concerning the Brownian loop measure in $\mathbb{R}^d$, $d \ge 3$. In particular, we derive decomposition formulas by rerooting the loops in specific ways and show that the loop measure is conformally invariant, generalising results of [Lup18] in dimension 1 and [LW04] in dimension 2. |
| title | Three-dimensional Brownian loop soup clusters |
| topic | Probability Mathematical Physics |
| url | https://arxiv.org/abs/2601.04840 |