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| Main Authors: | , |
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| Format: | Preprint |
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2022
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| Online Access: | https://arxiv.org/abs/2202.07558 |
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| _version_ | 1866909083870691328 |
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| author | Chang, Yinshan Zheng, Anqi |
| author_facet | Chang, Yinshan Zheng, Anqi |
| contents | Let $\{X_{v}:v\in\mathbb{Z}^d\}$ be i.i.d. random variables. Let $S(π)=\sum_{v\inπ}X_v$ be the weight of a self-avoiding lattice path $π$. Let \[M_n=\max\{S(π):π\text{ has length }n\text{ and starts from the origin}\}.\] We are interested in the asymptotics of $M_n$ as $n\to\infty$.
This model is closely related to the first passage percolation when the weights $\{X_v:v\in\mathbb{Z}^d\}$ are non-positive and it is closely related to the last passage percolation when the weights $\{X_v,v\in\mathbb{Z}^d\}$ are non-negative. For general weights, this model could be viewed as an interpolation between first passage models and last passage models. Besides, this model is also closely related to a variant of the position of right-most particles of branching random walks.
Under the two assumptions that $\existsα>0$, $E(X_0^{+})^d(\log^{+}X_0^{+})^{d+α}<+\infty$ and that $E[X_0^{-}]<+\infty$, we prove that there exists a finite real number $M$ such that $M_n/n$ converges to a deterministic constant $M$ in $L^{1}$ as $n$ tends to infinity. And under the stronger assumptions that $\existsα>0$, $E(X_0^{+})^d(\log^{+}X_0^{+})^{d+α}<+\infty$ and that $E[(X_0^{-})^4]<+\infty$, we prove that $M_n/n$ converges to the same constant $M$ almost surely as $n$ tends to infinity. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2202_07558 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Greedy lattice paths with general weights Chang, Yinshan Zheng, Anqi Probability 60K35 Let $\{X_{v}:v\in\mathbb{Z}^d\}$ be i.i.d. random variables. Let $S(π)=\sum_{v\inπ}X_v$ be the weight of a self-avoiding lattice path $π$. Let \[M_n=\max\{S(π):π\text{ has length }n\text{ and starts from the origin}\}.\] We are interested in the asymptotics of $M_n$ as $n\to\infty$. This model is closely related to the first passage percolation when the weights $\{X_v:v\in\mathbb{Z}^d\}$ are non-positive and it is closely related to the last passage percolation when the weights $\{X_v,v\in\mathbb{Z}^d\}$ are non-negative. For general weights, this model could be viewed as an interpolation between first passage models and last passage models. Besides, this model is also closely related to a variant of the position of right-most particles of branching random walks. Under the two assumptions that $\existsα>0$, $E(X_0^{+})^d(\log^{+}X_0^{+})^{d+α}<+\infty$ and that $E[X_0^{-}]<+\infty$, we prove that there exists a finite real number $M$ such that $M_n/n$ converges to a deterministic constant $M$ in $L^{1}$ as $n$ tends to infinity. And under the stronger assumptions that $\existsα>0$, $E(X_0^{+})^d(\log^{+}X_0^{+})^{d+α}<+\infty$ and that $E[(X_0^{-})^4]<+\infty$, we prove that $M_n/n$ converges to the same constant $M$ almost surely as $n$ tends to infinity. |
| title | Greedy lattice paths with general weights |
| topic | Probability 60K35 |
| url | https://arxiv.org/abs/2202.07558 |