  We consider directed last-passage percolation on the random graph G = (V,E)
where V = Z and each edge (i,j), for i < j, is present in E independently with
some probability 0 < p <= 1. To every present edge (i,j) we attach i.i.d.
random weights v_{i,j} > 0. We are interested in the behaviour of w_{0,n},
which is the maximum weight of all directed paths from 0 to n, as n tends to
infinity. We see two very different types of behaviour, depending on whether
E[v_{i,j}^2] is finite or infinite. In the case where E[v_{i,j}^2] is finite 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[v_{i,j}^2] is infinite 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 by
Hambly and Martin.
