We study collective behavior of locally-coupled limit-cycle oscillators withscattered intrinsic frequencies on $d$-dimensional lattices. A linear analysisshows that the system should be always desynchronized up to $d=4$. On the otherhand, numerical investigation for $d= 5$ and 6 reveals the emergence of thesynchronized (ordered) phase via a continuous transition from the fully randomdesynchronized phase. This demonstrates that the lower critical dimension forthe phase synchronization in this system is $d_{l}=4$
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机译：我们研究了在$ d $维网格上具有固有频率分散的局部耦合极限周期振荡器的集体行为。线性分析表明，系统应始终不同步至$ d = 4 $。另一方面，对$ d = 5 $和6的数值研究揭示了同步（有序）相通过从完全随机去同步相的连续跃迁而出现。这表明在此系统中，相位同步的较低临界尺寸为$ d_ {l} = 4 $
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