Sharp deviation bounds for midpoint and endpoint of geodesics in exponential last passage percolation
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| Format: | Preprint |
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2024
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| _version_ | 1866913684217921536 |
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| author | Agarwal, Pranay Basu, Riddhipratim |
| author_facet | Agarwal, Pranay Basu, Riddhipratim |
| contents | For exponential last passage percolation on the plane we analyse the probability that the point-to-line geodesic exhibits an atypically large transversal fluctuation at the endpoint as well as the probability that the point-to-point geodesic exhibits an atypically large transversal fluctuation at the halfway point. In particular, we show that $p^*_n(t)$, the probability that the point-to-line geodesic from the origin to the line $x+y=2n$ ends at $(n-t(2n)^{2/3}, n+t(2n)^{2/3})$ satisfies that $n^{2/3}p^*_n(t)=\exp(-(\frac{4}{3}+o(1))t^{3})$ for $t$ large and $p_{n,\frac{1}{2}}(t)$, the probability that the geodesic from the origin to the point $(n,n)$ passes through the point $(\frac{1}{2}n-tn^{2/3}, \frac{1}{2} n+tn^{2/3})$, satisfies $n^{2/3}p_{n,\frac{1}{2}}(t)=\exp(-(\frac{8}{3}+o(1))t^3)$ for $t$ large. The latter result solves a special case of a conjecture from Liu (PTRF, 2022). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_18056 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Sharp deviation bounds for midpoint and endpoint of geodesics in exponential last passage percolation Agarwal, Pranay Basu, Riddhipratim Probability For exponential last passage percolation on the plane we analyse the probability that the point-to-line geodesic exhibits an atypically large transversal fluctuation at the endpoint as well as the probability that the point-to-point geodesic exhibits an atypically large transversal fluctuation at the halfway point. In particular, we show that $p^*_n(t)$, the probability that the point-to-line geodesic from the origin to the line $x+y=2n$ ends at $(n-t(2n)^{2/3}, n+t(2n)^{2/3})$ satisfies that $n^{2/3}p^*_n(t)=\exp(-(\frac{4}{3}+o(1))t^{3})$ for $t$ large and $p_{n,\frac{1}{2}}(t)$, the probability that the geodesic from the origin to the point $(n,n)$ passes through the point $(\frac{1}{2}n-tn^{2/3}, \frac{1}{2} n+tn^{2/3})$, satisfies $n^{2/3}p_{n,\frac{1}{2}}(t)=\exp(-(\frac{8}{3}+o(1))t^3)$ for $t$ large. The latter result solves a special case of a conjecture from Liu (PTRF, 2022). |
| title | Sharp deviation bounds for midpoint and endpoint of geodesics in exponential last passage percolation |
| topic | Probability |
| url | https://arxiv.org/abs/2405.18056 |