  We construct a random MERA state with a bond dimension that varies with the
level of the MERA. This causes the state to exhibit a very different
entanglement structure from that usually seen in MERA, with neighboring
intervals of length $l$ exhibiting a mutual information proportional to
$\epsilon l$ for some constant $\epsilon$, up to a length scale exponentially
large in $\epsilon$. We express the entropy of a random MERA in terms of sums
over cuts through the MERA network, with the entropy in this case controlled by
the cut minimizing bond dimensions cut through. One motivation for this
construction is to investigate the tightness of the Brandao-Horodecki\cite{bh}
entropy bound relating entanglement to correlation decay. Using the random
MERA, we show that at least part of the proof is tight: there do exist states
with the required property of having linear mutual information between
neighboring intervals at all length scales. We conjecture that this state has
exponential correlation decay and that it demonstrates that the
Brandao-Horodecki bound is tight (at least up to constant factors), and we
provide some numerical evidence for this as well as a sketch of how a proof of
correlation decay might proceed.
