The height gap of planar Brownian motion is $\frac{5}π$
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866914378802003968 |
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| author | Jego, Antoine Lupu, Titus Qian, Wei |
| author_facet | Jego, Antoine Lupu, Titus Qian, Wei |
| contents | We show that the occupation measure of planar Brownian motion exhibits a constant height gap of $5/π$ across its outer boundary. This property bears similarities with the celebrated results of Schramm--Sheffield [18] and Miller--Sheffield [12] concerning the height gap of the Gaussian free field across SLE$_4$/CLE$_4$ curves. Heuristically, our result can also be thought of as the $θ\to 0^+$ limit of the height gap property of a field built out of a Brownian loop soup with subcritical intensity $θ>0$, proved in our recent paper [3]. To obtain the explicit value of the height gap, we rely on the computation by Garban and Trujillo Ferreras [1] of the expected area of the domain delimited by the outer boundary of a Brownian bridge. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_03040 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The height gap of planar Brownian motion is $\frac{5}π$ Jego, Antoine Lupu, Titus Qian, Wei Probability Mathematical Physics 60J65, 60J67, 60J55 We show that the occupation measure of planar Brownian motion exhibits a constant height gap of $5/π$ across its outer boundary. This property bears similarities with the celebrated results of Schramm--Sheffield [18] and Miller--Sheffield [12] concerning the height gap of the Gaussian free field across SLE$_4$/CLE$_4$ curves. Heuristically, our result can also be thought of as the $θ\to 0^+$ limit of the height gap property of a field built out of a Brownian loop soup with subcritical intensity $θ>0$, proved in our recent paper [3]. To obtain the explicit value of the height gap, we rely on the computation by Garban and Trujillo Ferreras [1] of the expected area of the domain delimited by the outer boundary of a Brownian bridge. |
| title | The height gap of planar Brownian motion is $\frac{5}π$ |
| topic | Probability Mathematical Physics 60J65, 60J67, 60J55 |
| url | https://arxiv.org/abs/2403.03040 |