Gaussian Broadcast on Grids

Motivated by the classical work on finite noisy automata (Gray 1982, G\'{a}cs 2001, Gray 2001) and by the recent work on broadcasting on grids (Makur, Mossel, and Polyanskiy 2022), we introduce Gaussian variants of these models. These models are defined on graded posets. At time $0$, all nodes begin with $X_0$. At time $k\ge 1$, each node on layer $k$ computes a combination of its inputs at layer $k-1$ with independent Gaussian noise added. When is it possible to recover $X_0$ 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 two models defined on grids: In the infinite model, layer $k$ is the vertices of $\mathbb{Z}^{d+1}$ whose sum of entries is $k$ and for a vertex $v$ at layer $k \ge 1$, $X_v=\alpha\sum (X_u + W_{u,v})$, summed over all $u$ on layer $k-1$ that differ from $v$ exactly in one coordinate, and $W_{u,v}$ are i.i.d. $\mathcal{N}(0,1)$. We show that when $\alpha<1/(d+1)$, the correlation between $X_v$ and $X_0$ decays exponentially, and when $\alpha>1/(d+1)$, the correlation is bounded away from $0$. The critical case when $\alpha=1/(d+1)$ exhibits a phase transition in dimension, where $X_v$ has non-vanishing correlation with $X_0$ if and only if $d\ge 3$. The same results hold for any bounded window. In the finite model, layer $k$ is the vertices of $\mathbb{Z}^{d+1}$ with nonnegative entries with sum $k$. We identify the sub-critical and the super-critical regimes. In the sub-critical regime, the correlation decays to $0$ for unbounded windows. In the super-critical regime, there exists for every $t$ a convex combination of $X_u$ on layer $t$ whose correlation is bounded away from $0$. We find that for the critical parameters, the correlation is vanishing in all dimensions and for unbounded window sizes.