Greedy Lattice Paths with General Weights

Let{X_(v):v∈Z^(d)}be i.i.d.random variables.Let S(π)=Σ_(v∈π)X_(v)be the weight of a self-avoiding lattice pathπ.Let M_(n)=max{S(π):πhas length n and starts from the origin}.We are interested in the asymptotics of Mn as n→∞.This model is closely related to the first passage percolation when the weights{X_(v):v∈Z^(d)}are non-positive and it is closely related to the last passage percolation when the weights{X_(v):v∈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∃α>0,E(X_(0)^(+))^(d)(log^(+)X_(0)^(+))^(d+α)<+∞and that E[X_(0)^(−)]<+∞,we prove that there exists a finite real number M such that Mn/n converges to a deterministic constant M in L^(1)as n tends to infinity.And under the stronger assumptions that∃α>0,E(X_(0)^(+))^(d)(log^(+)X_(0)^(+))^(d+α)<+∞and that E[(X_(0)^(−))^(4)]<+∞,we prove that M_(n)/n converges to the same constant M almost surely as n tends to infinity.