Gaussian broadcast on grids

Motivated by the classical work on finite noisy automata and by the recent work on broadcasting on grids, we introduce Gaussian variants of these models. These models are defined on graded posets. At time (Formula presented), all nodes begin with (Formula presented). At time (Formula presented), each node on layer (Formula presented) computes a combination of its inputs at layer (Formula presented) with independent Gaussian noise added. When is it possible to recover (Formula presented) with non-vanishing correlation? We consider different notions of recovery including recovery from a single node, recovery from a bounded window and recovery from an unbounded window. Our main interest is in the following two models defined on grids. In the infinite model, layer (Formula presented) is the vertices of (Formula presented) whose sum of entries is (Formula presented) and for a vertex (Formula presented) at layer (Formula presented), (Formula presented), summed over all (Formula presented) on layer (Formula presented) that differ from (Formula presented) exactly in one coordinate, and (Formula presented) are i.i.d. (Formula presented). We show that when (Formula presented), the correlation between (Formula presented) and (Formula presented) decays exponentially, and when (Formula presented), the correlation is bounded away from (Formula presented). The critical case when (Formula presented) exhibits a phase transition in dimension, where (Formula presented) has non-vanishing correlation with (Formula presented) if and only if (Formula presented). The same results hold for any bounded window. In the finite model, layer (Formula presented) is the vertices of (Formula presented) with non-negative entries with sum (Formula presented). We identify the sub-critical and the super-critical regimes. In the sub-critical regime, the correlation decays to (Formula presented) for unbounded windows. In the super-critical regime, there exists for every (Formula presented) a convex combination of (Formula presented) on layer (Formula presented) whose correlation is bounded away from (Formula presented). We find that for the critical parameters, the correlation is vanishing in all dimensions and for unbounded window sizes.