We study the role of the tail and the range of interaction in a spatially structured population of two-state on-off units governed by Markovian transition rates. The coupling among the oscillators is evidenced by the dependence of the transition rates of each unit on the states of the units to which it is coupled. Tuning the tail or range of the interactions, we observe a transition from an ordered global state (long-range interactions) to a disordered one (short-range interactions). Depending on the interaction kernel, the transition may be smooth (second order) or abrupt (first order). We analyze the transient, which may present different routes to the steady state with vastly different time scales.
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