Three-halves variation of geodesics in the directed landscape
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2020
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| _version_ | 1866929567649759232 |
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| author | Dauvergne, Duncan Sarkar, Sourav Virág, Bálint |
| author_facet | Dauvergne, Duncan Sarkar, Sourav Virág, Bálint |
| contents | We show that geodesics in the directed landscape have $3/2$-variation and that weight functions along the geodesics have cubic variation.
We show that the geodesic and its landscape environment around an interior point has a small-scale limit. This limit is given in terms of the directed landscape with Brownian-Bessel boundary conditions. The environments around different interior points are asymptotically independent.
We give tail bounds with optimal exponents for geodesic and weight function increments.
As an application of our results, we show that geodesics are not Hölder-$2/3$ and that weight functions are not Hölder-$1/3$, although these objects are known to be Hölder with all lower exponents. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2010_12994 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Three-halves variation of geodesics in the directed landscape Dauvergne, Duncan Sarkar, Sourav Virág, Bálint Probability 82B23, 82C22, 60H15 We show that geodesics in the directed landscape have $3/2$-variation and that weight functions along the geodesics have cubic variation. We show that the geodesic and its landscape environment around an interior point has a small-scale limit. This limit is given in terms of the directed landscape with Brownian-Bessel boundary conditions. The environments around different interior points are asymptotically independent. We give tail bounds with optimal exponents for geodesic and weight function increments. As an application of our results, we show that geodesics are not Hölder-$2/3$ and that weight functions are not Hölder-$1/3$, although these objects are known to be Hölder with all lower exponents. |
| title | Three-halves variation of geodesics in the directed landscape |
| topic | Probability 82B23, 82C22, 60H15 |
| url | https://arxiv.org/abs/2010.12994 |