  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.
