Geodesic switches and exceptional times in dynamical Brownian last passage percolation
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915588795793408 |
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| author | Bhatia, Manan |
| author_facet | Bhatia, Manan |
| contents | We consider Brownian last passage percolation evolving dynamically via a discrete resampling procedure. Using $Γ_{(0,0)}^{(n,n),r}$ to denote a geodesic from $(0,0)$ to $(n,n)$ at time $r$, we prove that the expected total number of coarse-grained changes (or "switches") accumulated by $Γ_{(0,0)}^{(n,n),r}$ away from its endpoints during a time interval $[s,t]$ is at most $n^{5/3+o(1)}(t-s)$; we expect the exponent $5/3$ to be tight. Using the above estimate, we establish that the set $\mathscr{T}$ of exceptional times at which a non-trivial bi-infinite geodesic exists a.s. has Hausdorff dimension at most $1/2$. Further, for any fixed direction $θ$, we show that the set $\mathscr{T}^θ\subseteq \mathscr{T}$ of times at which a non-trivial bi-infinite geodesic directed along $θ$ exists a.s. has Hausdorff dimension equal to $0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_27589 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Geodesic switches and exceptional times in dynamical Brownian last passage percolation Bhatia, Manan Probability We consider Brownian last passage percolation evolving dynamically via a discrete resampling procedure. Using $Γ_{(0,0)}^{(n,n),r}$ to denote a geodesic from $(0,0)$ to $(n,n)$ at time $r$, we prove that the expected total number of coarse-grained changes (or "switches") accumulated by $Γ_{(0,0)}^{(n,n),r}$ away from its endpoints during a time interval $[s,t]$ is at most $n^{5/3+o(1)}(t-s)$; we expect the exponent $5/3$ to be tight. Using the above estimate, we establish that the set $\mathscr{T}$ of exceptional times at which a non-trivial bi-infinite geodesic exists a.s. has Hausdorff dimension at most $1/2$. Further, for any fixed direction $θ$, we show that the set $\mathscr{T}^θ\subseteq \mathscr{T}$ of times at which a non-trivial bi-infinite geodesic directed along $θ$ exists a.s. has Hausdorff dimension equal to $0$. |
| title | Geodesic switches and exceptional times in dynamical Brownian last passage percolation |
| topic | Probability |
| url | https://arxiv.org/abs/2510.27589 |