355 research outputs found
Renyi entropies for classical stringnet models
In quantum mechanics, stringnet condensed states - a family of prototypical
states exhibiting non-trivial topological order - can be classified via their
long-range entanglement properties, in particular topological corrections to
the prevalent area law of the entanglement entropy. Here we consider classical
analogs of such stringnet models whose partition function is given by an
equal-weight superposition of classical stringnet configurations. Our analysis
of the Shannon and Renyi entropies for a bipartition of a given system reveals
that the prevalent volume law for these classical entropies is augmented by
subleading topological corrections that are intimately linked to the anyonic
theories underlying the construction of the classical models. We determine the
universal values of these topological corrections for a number of underlying
anyonic theories including su(2)_k, su(N)_1, and su(N)_2 theories
Ramanujan Coverings of Graphs
Let be a finite connected graph, and let be the spectral radius of
its universal cover. For example, if is -regular then
. We show that for every , there is an -covering
(a.k.a. an -lift) of where all the new eigenvalues are bounded from
above by . It follows that a bipartite Ramanujan graph has a Ramanujan
-covering for every . This generalizes the case due to Marcus,
Spielman and Srivastava (2013).
Every -covering of corresponds to a labeling of the edges of by
elements of the symmetric group . We generalize this notion to labeling
the edges by elements of various groups and present a broader scenario where
Ramanujan coverings are guaranteed to exist.
In particular, this shows the existence of richer families of bipartite
Ramanujan graphs than was known before. Inspired by Marcus-Spielman-Srivastava,
a crucial component of our proof is the existence of interlacing families of
polynomials for complex reflection groups. The core argument of this component
is taken from a recent paper of them (2015).
Another important ingredient of our proof is a new generalization of the
matching polynomial of a graph. We define the -th matching polynomial of
to be the average matching polynomial of all -coverings of . We show this
polynomial shares many properties with the original matching polynomial. For
example, it is real rooted with all its roots inside .Comment: 38 pages, 4 figures, journal version (minor changes from previous
arXiv version). Shortened version appeared in STOC 201
Isometric endomorphisms of free groups
An arbitrary homomorphism between groups is nonincreasing for stable
commutator length, and there are infinitely many (injective) homomorphisms
between free groups which strictly decrease the stable commutator length of
some elements. However, we show in this paper that a random homomorphism
between free groups is almost surely an isometry for stable commutator length
for every element; in particular, the unit ball in the scl norm of a free group
admits an enormous number of exotic isometries.
Using similar methods, we show that a random fatgraph in a free group is
extremal (i.e. is an absolute minimizer for relative Gromov norm) for its
boundary; this implies, for instance, that a random element of a free group
with commutator length at most n has commutator length exactly n and stable
commutator length exactly n-1/2. Our methods also let us construct explicit
(and computable) quasimorphisms which certify these facts.Comment: 26 pages, 6 figures; minor typographical edits for final published
versio
- …