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We consider $\mathbb{Z}_2$-synchronization on the Euclidean lattice. Everyvertex of $\mathbb{Z}^d$ is assigned an independent symmetric random sign$\theta_u$, and for every edge $(u,v)$ of the lattice, one observes the product$\theta_u\theta_v$ flipped independently with probability $p$. The task is toreconstruct products $\theta_u\theta_v$ for pairs of vertices $u$ and $v$ whichare arbitrarily far apart. Abb\'e, Massouli\'e, Montanari, Sly and Srivastava(2018) showed that synchronization is possible if and only if $p$ is below acritical threshold $\tilde{p}_c(d)$, and efficiently so for $p$ small enough.We augment this synchronization setting with a model of side informationpreserving the sign symmetry of $\theta$, and propose an \emph{efficient}algorithm which synchronizes a randomly chosen pair of far away vertices onaverage, up to a differently defined critical threshold $p_c(d)$. We conjecturethat $ p_c(d)=\tilde{p}_c(d)$ for all $d \ge 2$. Our strategy is to\emph{renormalize} the synchronization model in order to reduce the effectivenoise parameter, and then apply a variant of the multiscale algorithm of AMMSS.The success of the renormalization procedure is conditional on a plausible butunproved assumption about the regularity of the free energy of an Ising spinglass model on $\mathbb{Z}^d$.

Efficient $\mathbb{Z}_2$ synchronization on $\mathbb{Z}^d$ under symmetry-preserving side information