Consider the $d$-dimensional square lattice. We give each vertex a general vertex-weight, which forms i.i.d. random variables. For a self-avoiding lattice path, its path weight is defined to be the sum of weights of the vertices on this path. Let $M(n)$ be the maximal path-weight among all self-avoiding paths of length $n$ which start from the origin. Under certain assumptions, we prove that $M(n)$ grows linearly, i.e. we prove the a.s. and $L^{1}$ convergence of $M(n)/n$ as $n\to\infty$.

Joint work with Anqi Zheng.