Source: Ann. Appl. Probab.
Volume 13, Number 3
Consider a tree network $T$, where each edge acts as an independent copy of
a given channel $M$, and information is propagated from the root.
For which $T$ and $M$ does the configuration
obtained at level $n$ of $T$
typically contain significant information on the root variable?
This problem arose independently in biology, information theory and
For all $b$, we construct a channel
for which the variable at the root of the break $b$-ary tree
is independent of the configuration at the second level of that tree,
yet for sufficiently large $B>b$, the mutual information between
the configuration at level $n$ of
the $B$-ary tree and the root variable is bounded away from zero
for all $n$.
This construction is related to Reed--Solomon codes.
We improve the upper bounds on information flow
for asymmetric binary channels (which correspond to the Ising model
with an external field) and for symmetric $q$-ary channels
(which correspond to Potts models).
Let $\lam_2(M)$ denote the second largest
eigenvalue of $M$, in absolute value. A CLT of Kesten and
Stigum implies that if
$b |\lam_2(M)|^2 >1$, then the census
of the variables at any level of
the $b$-ary tree, contains significant information on the root variable.
We establish a converse: If $b |\lam_2(M)|^2 < 1$, then the
census of the variables at level $n$ of the $b$-ary tree is
asymptotically independent of the root variable.
This contrasts with examples where $b |\lam_2(M)|^2 <1$,
yet the configuration at level $n$
is not asymptotically independent of the root variable.
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