Nonequilibrium phase transition in a spreading process on a timeline

We consider a nonequilibrium process on a timeline with discrete sites which evolves by a non-Markovian update rule in such a way that an active site at time t activates one or several sites in the future at time t+dt. The time intervals dt are distributed algebraically as dt^(-1-kappa), where 0<kappa<1 is a control paramter. Depending on the activation rate, the system display a nonequilibrium phase transition which may be interpreted as directed percolation transition driven by temporal Levy flights in the limit of zero space dimensions. The critical properties are investigated by extensive numerical simulations and compared with field-theoretic predictions.