1,698 research outputs found
Invariance of KMS states on graph C*-algebras under classical and quantum symmetry
We study invariance of KMS states on graph C*-algebras coming from strongly
connected and circulant graphs under the classical and quantum symmetry of the
graphs. We show that the unique KMS state for strongly connected graphs is
invariant under quantum automorphism group of the graph. For circulant graphs,
it is shown that the action of classical and quantum automorphism group
preserves only one of the KMS states occurring at the critical inverse
temperature. We also give an example of a graph C*-algebra having more than one
KMS state such that all of them are invariant under the action of classical
automorphism group of the graph, but there is a unique KMS state which is
invariant under the action of quantum automorphism group of the graph.Comment: 15 pages, 2 figure
Good potentials for almost isomorphism of countable state Markov shifts
Almost isomorphism is an equivalence relation on countable state Markov
shifts which provides a strong version of Borel conjugacy; still, for mixing
SPR shifts, entropy is a complete invariant of almost isomorphism. In this
paper, we establish a class of potentials on countable state Markov shifts
whose thermodynamic formalism is respected by almost isomorphism
Combinatorics and Geometry of Transportation Polytopes: An Update
A transportation polytope consists of all multidimensional arrays or tables
of non-negative real numbers that satisfy certain sum conditions on subsets of
the entries. They arise naturally in optimization and statistics, and also have
interest for discrete mathematics because permutation matrices, latin squares,
and magic squares appear naturally as lattice points of these polytopes.
In this paper we survey advances on the understanding of the combinatorics
and geometry of these polyhedra and include some recent unpublished results on
the diameter of graphs of these polytopes. In particular, this is a thirty-year
update on the status of a list of open questions last visited in the 1984 book
by Yemelichev, Kovalev and Kravtsov and the 1986 survey paper of Vlach.Comment: 35 pages, 13 figure
The Vertex Reinforced Jump Process and a Random Schr\"odinger operator on finite graphs
We introduce a new exponential family of probability distributions, which can
be viewed as a multivariate generalization of the Inverse Gaussian
distribution. Considered as the potential of a random Schr\"odinger operator,
this exponential family is related to the random field that gives the mixing
measure of the Vertex Reinforced Jump Process (VRJP), and hence to the mixing
measure of the Edge Reinforced Random Walk (ERRW), the so-called magic formula.
In particular, it yields by direct computation the value of the normalizing
constants of these mixing measures, which solves a question raised by Diaconis.
The results of this paper are instrumental in [Sabot-Zeng,2015], where several
properties of the VRJP and the ERRW are proved, in particular a functional
central limit theorem in transient regimes, and recurrence of the 2-dimensional
ERRW.Comment: 15 page
Quantum automorphism groups of homogeneous graphs
Associated to a finite graph is its quantum automorphism group . The
main problem is to compute the Poincar\'e series of , meaning the series
whose coefficients are multiplicities of 1 into tensor
powers of the fundamental representation. In this paper we find a duality
between certain quantum groups and planar algebras, which leads to a planar
algebra formulation of the problem. Together with some other results, this
gives for all homogeneous graphs having 8 vertices or less.Comment: 30 page
Almost isomorphism for countable state Markov shifts
Countable state Markov shifts are a natural generalization of the well-known
subshifts of finite type. They are the subject of current research both for
their own sake and as models for smooth dynamical systems. In this paper, we
investigate their almost isomorphism and entropy conjugacy and obtain a
complete classification for the especially important class of strongly positive
recurrent Markov shifts. This gives a complete classification up to entropy
conjugacy of the natural extensions of smooth entropy expanding maps, including
all smooth interval maps with non-zero topological entropy
On Structural and Spectral Properties of Distance Magic Graphs
A graph is said to be distance magic if there is a bijection
from a vertex set of to the first natural numbers such that for
each vertex , its weight given by is constant, where
is an open neighborhood of a vertex . In this paper, we introduce the
concept of -distance magic labeling and establish the necessary and
sufficient condition for a graph to be distance magic. Additionally, we
introduce necessary and sufficient conditions for a connected regular graph to
exhibit distance magic properties in terms of the eigenvalues of its adjacency
and Laplacian matrices. Furthermore, we study the spectra of distance magic
graphs, focusing on singular distance magic graphs. Also, we show that the
number of distance magic labelings of a graph is, at most, the size of its
automorphism group
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