title: Finite-size scaling, dynamic fluctuations, and hyperscaling relation in the Kuramoto model.
abstract: 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 same given unimodal distribution g(ω). In the "random" case, frequencies are generated independently in accordance with g(ω), which gives rise to oscillator number fluctuation within any given frequency interval. In the "regular" case, the N frequencies are generated in a deterministic manner that minimizes the oscillator number fluctuations, leading to quasiuniformly 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 it 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.
authors.affiliation: Department of Physics and Research Institute of Physics and Chemistry, Chonbuk National University, Jeonju 561-756, Korea.
authors.forename: Hyunsuk
authors.initials: H
authors.lastname: Hong
authors.affiliation: Service de Physique de l'Etat Condensé, CEA-Saclay, CNRS UMR 3680, 91191 Gif-sur-Yvette, France.|Beijing Computational Science Research Center, Beijing 100084, China.
authors.forename: Hugues
authors.initials: H
authors.lastname: Chaté
authors.affiliation: Beijing Computational Science Research Center, Beijing 100084, China.|Department of Physics, Hong Kong Baptist University, Kowloon Tong, Hong Kong SAR, China.
authors.forename: Lei-Han
authors.initials: LH
authors.lastname: Tang
authors.affiliation: School of Physics and QUC, Korea Institute for Advanced Study, Seoul 130-722, Korea.
authors.forename: Hyunggyu
authors.initials: H
authors.lastname: Park