Many-Body Theory of Synchronization by Long-Range Interactions

Synchronization of coupled oscillators on a $d$-dimensional lattice with the power-law coupling $G(r) = g_0/r^α$ and randomly distributed intrinsic frequency is analyzed. A systematic perturbation theory is developed to calculate the order parameter profile and correlation functions in powers of $ε= α/d-1$. For $α\le d$, the system exhibits a sharp synchronization transition as described by the conventional mean-field theory. For $α> d$, the transition is smeared by the quenched disorder, and the macroscopic order parameter $\Avψ$ decays slowly with $g_0$ as $|\Avψ| \propto g_0^2$.

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