Finite-size scaling, dynamic fluctuations, and hyperscaling relation in the Kuramoto model

We revisit the Kuramoto model to explore the finite-size scaling (FSS) of the order parameter and its dynamic fluctuations near the onset of the synchronization transition, paying particular attention to effects induced by the randomness of the intrinsic frequencies of oscillators. For a population of size $N$, we study two ways of sampling the intrinsic frequencies according to the {\it same} given unimodal distribution $g(\omega )$. In the {\em random}' case, frequencies are generated independently in accordance with $g(\omega)$, which gives rise to oscillator number fluctuation within any given frequency interval. In the {\em regular}’ case, the $N$ frequencies are generated in a deterministic manner that minimizes the oscillator number fluctuations, leading to quasi-uniformly spaced frequencies in the population. We find that the two samplings yield substantially different finite-size properties with clearly distinct scaling exponents. Moreover, the hyperscaling relation between the order parameter and its fluctuations is valid in the regular case, but is violated in the random case. In this last case, a self-consistent mean-field theory that completely ignores dynamic fluctuations correctly predicts the FSS exponent of the order parameter but not its critical amplitude.