  Magnetic zigzag edges of graphene are considered as a basis for novel
spintronics devices despite the fact that no true long-range magnetic order is
possible in one dimension. We study the transverse and longitudinal
fluctuations of magnetic moments at zigzag edges of graphene from first
principles. We find a high value for the spin wave stiffness $D$ = 2100 meV
\AA$^2$ and a spin-collinear domain wall creation energy $E_dw$ = 114 meV
accompanied by low magnetic anisotropy. Above the crossover temperature $T_x
\approx$10 K the spin correlation length $\xi \propto T^{-1}$ limits the
long-range magnetic order to ~1 nm at 300 K while below $T_x$ it grows
exponentially with decreasing temperature. We discuss possible ways of
increasing the range of magnetic order and effects of edge roughness on it.
