From stability to chaos in last-passage percolation
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866910337138163712 |
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| author | Ahlberg, Daniel Deijfen, Maria Sfragara, Matteo |
| author_facet | Ahlberg, Daniel Deijfen, Maria Sfragara, Matteo |
| contents | We study the transition from stability to chaos in a dynamic last passage percolation model on $\mathbb{Z}^d$ with random weights at the vertices. Given an initial weight configuration at time $0$, we perturb the model over time in such a way that the weight configuration at time $t$ is obtained by resampling each weight independently with probability $t$. On the cube $[0,n]^d$, we study geodesics, that is, weight-maximizing up-right paths from $(0,0, \dots, 0)$ to $(n,n, \dots, n)$, and their passage time $T$. Under mild conditions on the weight distribution, we prove a phase transition between stability and chaos at $t \asymp \frac{1}{n}\mathrm{Var}(T)$. Indeed, as $n$ grows large, for small values of $t$, the passage times at time $0$ and time $t$ are highly correlated, while for large values of $t$, the geodesics become almost disjoint. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_11379 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | From stability to chaos in last-passage percolation Ahlberg, Daniel Deijfen, Maria Sfragara, Matteo Probability 60K35 We study the transition from stability to chaos in a dynamic last passage percolation model on $\mathbb{Z}^d$ with random weights at the vertices. Given an initial weight configuration at time $0$, we perturb the model over time in such a way that the weight configuration at time $t$ is obtained by resampling each weight independently with probability $t$. On the cube $[0,n]^d$, we study geodesics, that is, weight-maximizing up-right paths from $(0,0, \dots, 0)$ to $(n,n, \dots, n)$, and their passage time $T$. Under mild conditions on the weight distribution, we prove a phase transition between stability and chaos at $t \asymp \frac{1}{n}\mathrm{Var}(T)$. Indeed, as $n$ grows large, for small values of $t$, the passage times at time $0$ and time $t$ are highly correlated, while for large values of $t$, the geodesics become almost disjoint. |
| title | From stability to chaos in last-passage percolation |
| topic | Probability 60K35 |
| url | https://arxiv.org/abs/2302.11379 |