  Let $\{X_{v}:v\in\mathbb{Z}^d\}$ be i.i.d. random variables. Let
$S(\pi)=\sum_{v\in\pi}X_v$ be the weight of a self-avoiding lattice path $\pi$.
Let \[M_n=\max\{S(\pi):\pi\text{ has length }n\text{ and starts from the
origin}\}.\] 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 $\exists\alpha>0$,
$E(X_0^{+})^d(\log^{+}X_0^{+})^{d+\alpha}<+\infty$ 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 $\exists\alpha>0$,
$E(X_0^{+})^d(\log^{+}X_0^{+})^{d+\alpha}<+\infty$ 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.
