We study the almost periods of the eigenmodes of flat planar manifolds in the high energy limit. We prove in particular that the Gaussian Arithmetic Random Waves replicate almost identically at a scale at most \(\ell _{n}:= n^{-\frac{1}{2}}\operatorname {exp}\left( {\mathcal {N}}_n\right) \) , where \({\mathcal {N}}_n\) is the number of ways n can be written as a sum of two squares. It provides a qualitative interpretation of the full correlation phenomenon of the nodal length, which is known to happen at scales larger than \(\ell _{n}':= n^{-1/2}{\mathcal {N}}_{n}^{A}.\) We provide also a heuristic with a toy model pleading that the minimal scale of replication should be closer to \(\ell _{n}'\) than \(\ell _{n}.\)
