Hyunggyu Park, Korea Institute of Advanced Study
on Collective Synchronizations and Fluctuations

The synchronization transition of coupled random frequency oscillators is revisited. The Kuramoto model (global coupling) is shown to exhibit unusual sample-to-sample fluctuations dominant over temporal fluctuations, resulting in the finite-size-scaling exponent $\bar\nu=5/2$. This may be generalized to scale-free networks where more interesting finite-size scaling is observed.Simulations of locally coupled oscillators in $d$-dimensions reveal two types of frequency entrainment: mean-field behavior at $d>4$, and aggregation of compact synchronized domains in three and four dimensions. In the latter case, scaling arguments yield the finite-size-scaling exponent $\nu=2/(d-2)$, in good agreement with numerical results. Finally we discuss over temporal fluctuations and hyperscaling relations in various mean-field environments