Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Broadcasting on Two-Dimensional Regular Grids   We study a specialization of the problem of broadcasting on directed acyclic
graphs, namely, broadcasting on 2D regular grids. Consider a 2D regular grid
with source vertex $X$ at layer $0$ and $k+1$ vertices at layer $k\geq 1$,
which are at distance $k$ from $X$. Every vertex of the 2D regular grid has
outdegree $2$, the vertices at the boundary have indegree $1$, and all other
vertices have indegree $2$. At time $0$, $X$ is given a 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 binary symmetric channels with
noise level $\delta\in(0,1/2)$. Then, each vertex combines its received bits
using a common Boolean processing function to produce an output bit. The
objective is to recover $X$ with probability of error better than $1/2$ from
all vertices at layer $k$ as $k \rightarrow \infty$. Besides their natural
interpretation in communication networks, such broadcasting processes can be
construed as 1D probabilistic cellular automata (PCA) with boundary conditions
that limit the number of sites at each time $k$ to $k+1$. We conjecture that it
is impossible to propagate information in a 2D regular grid regardless of the
noise level and the choice of processing function. In this paper, we make
progress towards establishing this conjecture, and prove using ideas from
percolation and coding theory that recovery of $X$ is impossible for any
$\delta$ provided that all vertices use either AND or XOR processing functions.
Furthermore, we propose a martingale-based approach that establishes the
impossibility of recovering $X$ for any $\delta$ when all NAND processing
functions are used if certain supermartingales can be rigorously constructed.
We also provide numerical evidence for the existence of these supermartingales
by computing explicit examples for different values of $\delta$ via linear
programming.

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The categories of this paper are: atom-ph
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The categories of this paper are: chem-ph
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The categories of this paper are: funct-an
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The categories of this paper are: supr-con
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The categories of this paper are: hep-lat
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The categories of this paper are: q-alg
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The categories of this paper are: nucl-ex
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The categories of this paper are: cond-mat
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The categories of this paper are: nlin
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The categories of this paper are: cmp-lg
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The categories of this paper are: mtrl-th
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The categories of this paper are: hep-th
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The categories of this paper are: q-fin
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The categories of this paper are: chao-dyn
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The categories of this paper are: hep-ph
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The categories of this paper are: econ