论文标题
一般重量的贪婪的格子路径
Greedy lattice paths with general weights
论文作者
论文摘要
令$ \ {x_ {v}：v \ in \ mathbb {z}^d \} $为i.i.d.随机变量。令$ s（π）= \ sum_ {v \inπ} x_v $是自我避免晶格路径$π$的重量。令\ [m_n = \ max \ {s（π）：π\ text {具有长度} n \ text {并从Origin} \}开始。 当权重$ \ {x_v：v \ in \ mathbb {z}^d \} $是无阳性时，该模型与第一个段落的渗透密切相关，并且与最后一段段落的percolation无阳性密切相关。对于一般权重，可以将该模型视为第一段模型与最后一段模型之间的插值。此外，该模型也与分支随机步行的最直接颗粒的位置的变体密切相关。 在两个假设下，$ \cestistα> 0 $，$ e（x_0^{+}）^d（\ log^{+} x_0^{+} {+}）^{d+α} <+\ \ \ \\ infty $ and $ e [x_0^{ - }] $ n $的确定性常数$ m $ in $ l^{1} $倾向于无限。并在更强有力的假设下，$ \存在$ 0 $，$ e（x_0^{+}）^d（\ log^{+} x_0^{+} {+}）^{d+α} <+α} <+\ \\ infty $ and $ e [（x_0^{ - }）肯定是$ n $倾向于无限。
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.
