Finite-size scaling at the Kuramoto synchronization transition
In plain words
In a finite population of oscillators the sharp synchronization transition is smeared, and how the smearing shrinks with population size reveals its universality class. Different ways of choosing the oscillator frequencies give different answers.
Precise statement
For the globally coupled Kuramoto model with $N$ oscillators and unimodal frequency distribution $g(\omega)$, determine the finite-size scaling exponent $\nu_{\mathrm{bar}}$ in $r(K, N) = N^{-\beta/\nu_{\mathrm{bar}}} F((K - K_c) N^{1/\nu_{\mathrm{bar}}})$, and its dependence on random versus deterministic sampling of frequencies and on added noise. An answer is $\nu_{\mathrm{bar}}$ for each case with an analytic derivation.
What would settle it
An analytic derivation of $\nu_{\mathrm{bar}}$ for each sampling scheme confirmed by simulations at $N$ up to $10^{6}$ or more.
Status in the literature
Numerics give $\nu_{\mathrm{bar}}$ near $5/2$ for random frequency sampling and a different value for regular sampling; an analytic derivation is incomplete.
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