Near-existence of bigeodesics in dynamical exponential last passage percolation
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911241829613568 |
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| author | Bhatia, Manan |
| author_facet | Bhatia, Manan |
| contents | It is believed that, under very general conditions, bi-infinite geodesics (or bigeodesics) do not exist for planar first and last passage percolation (LPP) models. However, if one endows the model with a natural dynamics, thereby gradually perturbing the geometry, then it is plausible that there could exist a non-trivial set $\mathscr{T}$ of exceptional times at which such bigeodesics exist. For dynamical exponential LPP, we show that $\mathscr{T}$ is "very close" to being non-trivial; namely, we obtain an $Ω( 1/\log n)$ lower bound on the probability that there exists a random time $t\in [0,1]$ at which a non-trivial geodesic of length $n$ passes through the origin at its midpoint; note that if the above probability were $Ω(1)$, then it would imply the non-triviality of $\mathscr{T}$. We conjecture that, even if $\mathscr{T}\neq \emptyset$, it a.s. has Hausdorff dimension exactly zero. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_12293 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Near-existence of bigeodesics in dynamical exponential last passage percolation Bhatia, Manan Probability It is believed that, under very general conditions, bi-infinite geodesics (or bigeodesics) do not exist for planar first and last passage percolation (LPP) models. However, if one endows the model with a natural dynamics, thereby gradually perturbing the geometry, then it is plausible that there could exist a non-trivial set $\mathscr{T}$ of exceptional times at which such bigeodesics exist. For dynamical exponential LPP, we show that $\mathscr{T}$ is "very close" to being non-trivial; namely, we obtain an $Ω( 1/\log n)$ lower bound on the probability that there exists a random time $t\in [0,1]$ at which a non-trivial geodesic of length $n$ passes through the origin at its midpoint; note that if the above probability were $Ω(1)$, then it would imply the non-triviality of $\mathscr{T}$. We conjecture that, even if $\mathscr{T}\neq \emptyset$, it a.s. has Hausdorff dimension exactly zero. |
| title | Near-existence of bigeodesics in dynamical exponential last passage percolation |
| topic | Probability |
| url | https://arxiv.org/abs/2504.12293 |