  Motivated by long-range dispersal in ecological systems, we formulate and
apply a general strong-disorder renormalization group (SDRG) framework to
describe one-dimensional disordered contact processes with heavy-tailed, such
as power law, stretched exponential, and log-normal dispersal kernels, widely
used in ecology. The focus is on the close-to-critical scaling of the order
parameters, including the commonly used density, as well as the less known
persistence, which is non-zero in the inactive phase. Our analytic and
numerical results obtained by SDRG schemes at different levels of approximation
reveal that the more slowly decaying dispersal kernels lead to more smoothly
vanishing densities as the critical point is approached. The persistence,
however, shows an opposite tendency: the broadening of the dispersal makes its
decline more singular at the critical point, becoming discontinuous for the
extreme case of power-law dispersal. The SDRG schemes presented here also
describe the quantum phase transition of random transverse-field Ising chains
with ferromagnetic long-range interactions, the density corresponding to the
magnetization of this model.
