Atypical stars on a directed landscape geodesic
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866916188075851776 |
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| author | Bhatia, Manan |
| author_facet | Bhatia, Manan |
| contents | In random geometry, a recurring theme is that any two geodesics emanating from a typical point part ways at a strictly positive distance from the above point, and we call such points as $1$-stars. However, the measure zero set of atypical stars, the points where such coalescence fails, is typically uncountable and the corresponding Hausdorff dimensions of these sets have been heavily investigated for a variety of models including the directed landscape, Liouville quantum gravity and the Brownian map. In this paper, we consider the directed landscape -- the scaling limit of last passage percolation as constructed in the work Dauvergne-Ortmann-Virág '18 -- and look into the Hausdorff dimension of the set of atypical stars lying on a geodesic. We show that the above dimension is almost surely equal to $1/3$. This is in contrast to Ganguly-Zhang '22, where it was shown that set of atypical stars on the line $\{x=0\}$ has dimension $2/3$. This reduction of the dimension from $2/3$ to $1/3$ yields a quantitative manifestation of the smoothing of the environment around a geodesic with regard to exceptional behaviour. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2211_05734 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Atypical stars on a directed landscape geodesic Bhatia, Manan Probability In random geometry, a recurring theme is that any two geodesics emanating from a typical point part ways at a strictly positive distance from the above point, and we call such points as $1$-stars. However, the measure zero set of atypical stars, the points where such coalescence fails, is typically uncountable and the corresponding Hausdorff dimensions of these sets have been heavily investigated for a variety of models including the directed landscape, Liouville quantum gravity and the Brownian map. In this paper, we consider the directed landscape -- the scaling limit of last passage percolation as constructed in the work Dauvergne-Ortmann-Virág '18 -- and look into the Hausdorff dimension of the set of atypical stars lying on a geodesic. We show that the above dimension is almost surely equal to $1/3$. This is in contrast to Ganguly-Zhang '22, where it was shown that set of atypical stars on the line $\{x=0\}$ has dimension $2/3$. This reduction of the dimension from $2/3$ to $1/3$ yields a quantitative manifestation of the smoothing of the environment around a geodesic with regard to exceptional behaviour. |
| title | Atypical stars on a directed landscape geodesic |
| topic | Probability |
| url | https://arxiv.org/abs/2211.05734 |