Broadcasting on Two-Dimensional Regular Grids

We study an important specialization of the general problem of broadcasting on directed acyclic graphs, namely, that of broadcasting on two-dimensional (2D) regular grids. Consider an infinite directed acyclic graph with the form of a 2D regular grid, which has a single source vertex
$X$
at layer 0, and
$k + 1$
vertices at layer
$k \geq 1$
, which are at a distance of
$k$
from
$X$
. Every vertex of the 2D regular grid has outdegree 2, the vertices at the boundary have indegree 1, and all other non-source vertices have indegree 2. At time 0,
$X$
is given a uniform random bit. At time
$k \geq 1$
, each vertex in layer
$k$
receives transmitted bits from its parents in layer
$k-1$
, where the bits pass through independent binary symmetric channels with common crossover probability
$\delta \in \left({0,\frac {1}{2}}\right)$
during the process of transmission. Then, each vertex at layer
$k$
with indegree 2 combines its two input bits using a common deterministic Boolean processing function to produce a single output bit at the vertex. The objective is to recover
$X$
with probability of error better than
$\frac {1}{2}$
from all vertices at layer
$k$
as
$k \rightarrow \infty $
. Besides their natural interpretation in the context of communication networks, such broadcasting processes can be construed as one-dimensional (1D) probabilistic cellular automata, or discrete-time statistical mechanical spin-flip systems on 1D lattices, with boundary conditions that limit the number of sites at each time
$k$
to
$k+1$
. Inspired by the literature surrounding the “positive rates conjecture” for 1D probabilistic cellular automata, we conjecture that it is impossible to propagate information in a 2D regular grid regardless of the noise level
$\delta $
and the choice of common Boolean processing function. In this paper, we make considerable progress towards establishing this conjecture, and prove using ideas from percolation and coding theory that recovery of
$X$
is impossible for any
$\delta \in \left({0,\frac {1}{2}}\right)$
provided that all vertices with indegree 2 use either AND or XOR for their processing functions. Furthermore, we propose a detailed and general martingale-based approach that establishes the impossibility of recovering
$X$
for any
$\delta \in \left({0,\frac {1}{2}}\right)$
when all NAND processing functions are used if certain structured supermartingales can be rigorously constructed. We also provide strong numerical evidence for the existence of these supermartingales by computing several explicit examples for different values of
$\delta $
via linear programming.