Disconnection probability of Brownian motion on an annulus
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866914042809942016 |
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| author | Cai, Gefei Fu, Xuesong Sun, Xin Xie, Zhuoyan |
| author_facet | Cai, Gefei Fu, Xuesong Sun, Xin Xie, Zhuoyan |
| contents | We derive an exact formula for the probability that a Brownian path on an annulus does not disconnect the two boundary components of the annulus. The leading asymptotic behavior of this probability is governed by the disconnection exponent obtained by Lawler-Schramm-Werner (2001) using the connection to Schramm-Loewner evolution (SLE). The derivation of our formula is based on this connection and the coupling with Liouville quantum gravity (LQG). As byproducts of our proof, we obtain a precise relation between Brownian motion on a disk stopped upon hitting the boundary and the SLE$_{8/3}$ loop measure on the disk; we also obtain a detailed description of the LQG surfaces cut by the outer boundary of stopped Brownian motion on a $\sqrt{8/3}$-LQG disk. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_14073 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Disconnection probability of Brownian motion on an annulus Cai, Gefei Fu, Xuesong Sun, Xin Xie, Zhuoyan Probability Mathematical Physics 60J65, 60J67, 60D05 We derive an exact formula for the probability that a Brownian path on an annulus does not disconnect the two boundary components of the annulus. The leading asymptotic behavior of this probability is governed by the disconnection exponent obtained by Lawler-Schramm-Werner (2001) using the connection to Schramm-Loewner evolution (SLE). The derivation of our formula is based on this connection and the coupling with Liouville quantum gravity (LQG). As byproducts of our proof, we obtain a precise relation between Brownian motion on a disk stopped upon hitting the boundary and the SLE$_{8/3}$ loop measure on the disk; we also obtain a detailed description of the LQG surfaces cut by the outer boundary of stopped Brownian motion on a $\sqrt{8/3}$-LQG disk. |
| title | Disconnection probability of Brownian motion on an annulus |
| topic | Probability Mathematical Physics 60J65, 60J67, 60D05 |
| url | https://arxiv.org/abs/2509.14073 |