  We examine diffusion-limited aggregation generated by a random walk on Z with
long jumps. We derive upper and lower bounds on the growth rate of the
aggregate as a function of the number moments a single step of the walk has.
Under various regularity conditions on the tail of the step distribution, we
prove that the diameter grows as n^{beta+o(1)}, with an explicitly given beta.
The growth rate of the aggregate is shown to have three phase transitions, when
the walk steps have finite third moment, finite variance, and, conjecturally,
finite half moment.
