  Synchronization in a lattice of a finite population of phase oscillators with
algebraically decaying, non-normalized coupling is studied by numerical
simulations. A critical level of decay is found, below which full locking takes
place if the population contains a sufficiently large number of elements. For
large number of oscillators and small coupling constant, numerical simulations
and analytical arguments indicate that a phase transition separating
synchronization from incoherence appears at a decay exponent value equal to the
number of dimensions of the lattice. In contrast with earlier results on
similar systems with normalized coupling, we have indication that for the decay
exponent less than the dimensions of the lattice and for large populations,
synchronization is possible even if the coupling is arbitarily weak. This
finding suggests that in organisms interacting through slowly decaying signals
like light or sound, collective oscillations can always be established if the
population is sufficiently large.
