Revision 1

Accepted · first revision · record

Title: Lower critical dimensions for synchronization on lattices
Plain: 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: 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.
Settled by: 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.
Kind: well-posed
Answer type: value
References: (empty)
Literature status: contested
Status note: (empty)
Posed since: 1987