We study epidemic processes with immunization on very large 1-dimensional
lattices, where at least some of the infections are non-local, with rates
decaying as power laws p(x) ~ x^{-sigma-1} for large distances x. When starting
with a single infected site, the cluster of infected sites stays always bounded
if σ>1 (and dies with probability 1, of its size is allowed to
fluctuate down to zero), but the process can lead to an infinite epidemic for
sigma <1. For sigma <0 the behavior is essentially of mean field type, but for
0 < sigma <= 1 the behavior is non-trivial, both for the critical and for
supercritical cases. For critical epidemics we confirm a previous prediction
that the critical exponents controlling the correlation time and the
correlation length are simply related to each other, and we verify detailed
field theoretic predictions for sigma --> 1/3. For sigma = 1 we find generic
power laws with continuously varying exponents even in the supercritical case,
and confirm in detail the predicted Kosterlitz-Thouless nature of the
transition. Finally, the mass N(t) of supercritical clusters seems to grow for
0 < sigma < 1 like a stretched exponential. The latter implies that networks
embedded in 1-d space with power-behaved link distributions have infinite
intrinsic dimension (based on the graph distance), but are not small world.Comment: 16 pages, including 28 figures; minor changes from version v
