1,035 research outputs found
Complexity is Simple
In this note we investigate the role of Lloyd's computational bound in
holographic complexity. Our goal is to translate the assumptions behind Lloyd's
proof into the bulk language. In particular, we discuss the distinction between
orthogonalizing and `simple' gates and argue that these notions are useful for
diagnosing holographic complexity. We show that large black holes constructed
from series circuits necessarily employ simple gates, and thus do not satisfy
Lloyd's assumptions. We also estimate the degree of parallel processing
required in this case for elementary gates to orthogonalize. Finally, we show
that for small black holes at fixed chemical potential, the orthogonalization
condition is satisfied near the phase transition, supporting a possible
argument for the Weak Gravity Conjecture first advocated in Brown et al
Processes on Unimodular Random Networks
We investigate unimodular random networks. Our motivations include their
characterization via reversibility of an associated random walk and their
similarities to unimodular quasi-transitive graphs. We extend various theorems
concerning random walks, percolation, spanning forests, and amenability from
the known context of unimodular quasi-transitive graphs to the more general
context of unimodular random networks. We give properties of a trace associated
to unimodular random networks with applications to stochastic comparison of
continuous-time random walk.Comment: 66 pages; 3rd version corrects formula (4.4) -- the published version
is incorrect --, as well as a minor error in the proof of Proposition 4.10;
4th version corrects proof of Proposition 7.1; 5th version corrects proof of
Theorem 5.1; 6th version makes a few more minor correction
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