Lower critical dimensions for synchronization on lattices
In plain words
When each oscillator is coupled only to its neighbors on a lattice and natural frequencies are random, synchronization may be impossible in low dimensions. The dimensions above which frequency locking and full phase locking can occur are not established.
Precise statement
For Kuramoto oscillators on a $d$-dimensional hypercubic lattice with nearest-neighbor coupling $K$ and i.i.d. natural frequencies of finite variance, determine the lower critical dimension $d_f$ for macroscopic frequency entrainment and $d_p$ for phase synchronization (nonzero global order parameter) at finite $K$ as $N \to \infty$. Fluctuation arguments suggest $d_f = 2 \text{ and } d_p = 4$; numerical evidence is mixed. An answer is the pair $(d_f, d_p)$ with proof or controlled numerics.
What would settle it
A proof of the absence or presence of entrainment and phase order in each $d$, or simulations with finite-size scaling in $d = 2\ \text{to}\ 5$ that fix both dimensions.