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

We consider $\mathbb{Z}_2$-synchronization on the Euclidean lattice. Every vertex 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 to reconstruct products $\theta_u\theta_v$ for pairs of vertices $u$ and $v$ which are arbitrarily far apart. Abb\é, Massouli\é, Montanari, Sly and Srivastava (2018) showed that synchronization is possible if and only if $p$ is below a critical threshold $\tilde{p}_c(d)$, and efficiently so for $p$ small enough. We augment this synchronization setting with a model of side information preserving the sign symmetry of $\theta$, and propose an \emph{efficient} algorithm which synchronizes a randomly chosen pair of far away vertices on average, up to a differently defined critical threshold $p_c(d)$. We conjecture that $p_c(d)=\tilde{p}_c(d)$ for all $d \ge 2$. Our strategy is to \emph{renormalize} the synchronization model in order to reduce the effective noise parameter, and then apply a variant of the multiscale algorithm of AMMSS. The success of the renormalization procedure is conditional on a plausible but unproved assumption about the regularity of the free energy of an Ising spin glass model on $\mathbb{Z}^d$.