Crossing exponent in the Brownian loop soup
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866917079662198784 |
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| author | Jego, Antoine Lupu, Titus Qian, Wei |
| author_facet | Jego, Antoine Lupu, Titus Qian, Wei |
| contents | We study the clusters of loops in a Brownian loop soup in some bounded two-dimensional domain with subcritical intensity $θ\in (0,1/2]$. We obtain an exact expression for the asymptotic probability of the existence of a cluster crossing a given annulus of radii $r$ and $r^s$ as $r \to 0$ ($s >1$ fixed). Relying on this result, we then show that the probability for a macroscopic cluster to hit a given disc of radius $r$ decays like $|\log r|^{-1+θ+ o(1)}$ as $r \to 0$. Finally, we characterise the polar sets of clusters, i.e. sets that are not hit by the closure of any cluster, in terms of $\log^α$-capacity.
This paper reveals a connection between the 1D and 2D Brownian loop soups. This connection in turn implies the existence of a second critical intensity $θ= 1$ that describes a phase transition in the percolative behaviour of large loops on a logarithmic scale targeting an interior point of the domain. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_03782 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Crossing exponent in the Brownian loop soup Jego, Antoine Lupu, Titus Qian, Wei Probability Mathematical Physics We study the clusters of loops in a Brownian loop soup in some bounded two-dimensional domain with subcritical intensity $θ\in (0,1/2]$. We obtain an exact expression for the asymptotic probability of the existence of a cluster crossing a given annulus of radii $r$ and $r^s$ as $r \to 0$ ($s >1$ fixed). Relying on this result, we then show that the probability for a macroscopic cluster to hit a given disc of radius $r$ decays like $|\log r|^{-1+θ+ o(1)}$ as $r \to 0$. Finally, we characterise the polar sets of clusters, i.e. sets that are not hit by the closure of any cluster, in terms of $\log^α$-capacity. This paper reveals a connection between the 1D and 2D Brownian loop soups. This connection in turn implies the existence of a second critical intensity $θ= 1$ that describes a phase transition in the percolative behaviour of large loops on a logarithmic scale targeting an interior point of the domain. |
| title | Crossing exponent in the Brownian loop soup |
| topic | Probability Mathematical Physics |
| url | https://arxiv.org/abs/2303.03782 |