  We are interested in the threshold phenomena for propagation in nonlocal
diffusion equations with some compactly supported initial data. In the
so-called bistable and ignition cases, we provide the first quantitative
estimates for such phenomena. The outcomes dramatically depend on the tails of
the dispersal kernel and can take a large variety of different forms. The
strategy is to combine sharp estimates of the tails of the sum of i.i.d. random
variables (coming, in particular, from large deviation theory) and the
construction of accurate sub-and super-solutions.
