An information-percolation bound for spin synchronization on general graphs

Abstract

This paper considers the problem of reconstructing $n$ independent uniform spins $X_{1},\dots,X_{n}$ living on the vertices of an $n$-vertex graph $G$, by observing their interactions on the edges of the graph. This captures instances of models such as (i) broadcasting on trees, (ii) block models, (iii) synchronization on grids, (iv) spiked Wigner models. The paper gives an upper bound on the mutual information between two vertices in terms of a bond percolation estimate. Namely, the information between two vertices’ spins is bounded by the probability that these vertices are connected when edges are opened with a probability that “emulates” the edge-information. Both the information and the open-probability are based on the Chi-squared mutual information. The main results allow us to re-derive known results for information-theoretic nonreconstruction in models (i)–(iv), with more direct or improved bounds in some cases, and to obtain new results, such as for a spiked Wigner model on grids. The main result also implies a new subadditivity property for the Chi-squared mutual information for symmetric channels and general graphs, extending the subadditivity property obtained by Evans–Kenyon–Peres–Schulman (Ann. Appl. Probab. 10 (2000) 410–433) for trees. Some cases of nonsymmetrical channels are also discussed.