Long-range last-passage percolation on the line

We consider directed last-passage percolation on the random graph G=(V,E) where V=Z and each edge (i,j), for i<j.Z, is present in E independently with some probability p.(0,1]. To every (i,j).E we attach i.i.d. random weights vi,j>0. We are interested in the behaviour of w0,n, which is the maximum weight of all directed paths from 0 to n, as n... We see two very different types of behaviour, depending on whether E[v2i,j]<. or E[v2i,j]=.. In the case where E[v2i,j]<. we show that the process has a certain regenerative structure, and prove a strong law of large numbers and, under an extra assumption, a functional central limit theorem. In the situation where E[v2i,j]=. we obtain scaling laws and asymptotic distributions expressed in terms of a .continuous last-passage percolation. model on [0,1]; these are related to corresponding results for two-dimensional last-passage percolation with heavy-tailed weights obtained in Hambly and Martin [Probab. Theory Related Fields 137 (2007) 227.275].