68,906 research outputs found
KMS states on the C*-algebras of reducible graphs
We consider the dynamics on the C*-algebras of finite graphs obtained by
lifting the gauge action to an action of the real line. Enomoto, Fujii and
Watatani proved that if the vertex matrix of the graph is irreducible, then the
dynamics on the graph algebra admits a single KMS state. We have previously
studied the dynamics on the Toeplitz algebra, and explicitly described a
finite-dimensional simplex of KMS states for inverse temperatures above a
critical value. Here we study the KMS states for graphs with reducible vertex
matrix, and for inverse temperatures at and below the critical value. We prove
a general result which describes all the KMS states at a fixed inverse
temperature, and then apply this theorem to a variety of examples. We find that
there can be many patterns of phase transition, depending on the behaviour of
paths in the underlying graph.Comment: Publication version. To appear in Ergodic Theory and Dynamical
System
Fully Dynamic Single-Source Reachability in Practice: An Experimental Study
Given a directed graph and a source vertex, the fully dynamic single-source
reachability problem is to maintain the set of vertices that are reachable from
the given vertex, subject to edge deletions and insertions. It is one of the
most fundamental problems on graphs and appears directly or indirectly in many
and varied applications. While there has been theoretical work on this problem,
showing both linear conditional lower bounds for the fully dynamic problem and
insertions-only and deletions-only upper bounds beating these conditional lower
bounds, there has been no experimental study that compares the performance of
fully dynamic reachability algorithms in practice. Previous experimental
studies in this area concentrated only on the more general all-pairs
reachability or transitive closure problem and did not use real-world dynamic
graphs.
In this paper, we bridge this gap by empirically studying an extensive set of
algorithms for the single-source reachability problem in the fully dynamic
setting. In particular, we design several fully dynamic variants of well-known
approaches to obtain and maintain reachability information with respect to a
distinguished source. Moreover, we extend the existing insertions-only or
deletions-only upper bounds into fully dynamic algorithms. Even though the
worst-case time per operation of all the fully dynamic algorithms we evaluate
is at least linear in the number of edges in the graph (as is to be expected
given the conditional lower bounds) we show in our extensive experimental
evaluation that their performance differs greatly, both on generated as well as
on real-world instances
Loop corrections in spin models through density consistency
Computing marginal distributions of discrete or semidiscrete Markov random
fields (MRFs) is a fundamental, generally intractable problem with a vast
number of applications in virtually all fields of science. We present a new
family of computational schemes to approximately calculate the marginals of
discrete MRFs. This method shares some desirable properties with belief
propagation, in particular, providing exact marginals on acyclic graphs, but it
differs with the latter in that it includes some loop corrections; i.e., it
takes into account correlations coming from all cycles in the factor graph. It
is also similar to the adaptive Thouless-Anderson-Palmer method, but it differs
with the latter in that the consistency is not on the first two moments of the
distribution but rather on the value of its density on a subset of values. The
results on finite-dimensional Isinglike models show a significant improvement
with respect to the Bethe-Peierls (tree) approximation in all cases and with
respect to the plaquette cluster variational method approximation in many
cases. In particular, for the critical inverse temperature of the
homogeneous hypercubic lattice, the expansion of
around of the proposed scheme is exact up to the order,
whereas the two latter are exact only up to the order.Comment: 12 pages, 3 figures, 1 tabl
Dynamic message-passing approach for kinetic spin models with reversible dynamics
A method to approximately close the dynamic cavity equations for synchronous
reversible dynamics on a locally tree-like topology is presented. The method
builds on a graph expansion to eliminate loops from the normalizations of
each step in the dynamics, and an assumption that a set of auxilary
probability distributions on histories of pairs of spins mainly have
dependencies that are local in time. The closure is then effectuated by
projecting these probability distributions on -step Markov processes. The
method is shown in detail on the level of ordinary Markov processes (),
and outlined for higher-order approximations (). Numerical validations of
the technique are provided for the reconstruction of the transient and
equilibrium dynamics of the kinetic Ising model on a random graph with
arbitrary connectivity symmetry.Comment: 6 pages, 4 figure
On the AdS Higher Spin / O(N) Vector Model Correspondence: degeneracy of the holographic image
We explore the conjectured duality between the critical O(N) vector model and
minimal bosonic massless higher spin (HS) theory in AdS. In the boundary free
theory, the conformal partial wave expansion (CPWE) of the four-point function
of the scalar singlet bilinear is reorganized to make it explicitly
crossing-symmetric and closed in the singlet sector, dual to the bulk HS gauge
fields. We are able to analytically establish the factorized form of the fusion
coefficients as well as the two-point function coefficient of the HS currents.
We insist in directly computing the free correlators from bulk graphs with the
unconventional branch. The three-point function of the scalar bilinear turns
out to be an "extremal" one at d=3. The four-leg bulk exchange graph can be
precisely related to the CPWs of the boundary dual scalar and its shadow. The
flow in the IR by Legendre transforming at leading 1/N, following the pattern
of double-trace deformations, and the assumption of degeneracy of the hologram
lead to the CPWE of the scalar four-point function at IR. Here we confirm some
previous results, obtained from more involved computations of skeleton graphs,
as well as extend some of them from d=3 to generic dimension 2<d<4.Comment: 22 pages, 5 figure
Finitely ramified iterated extensions
Let K be a number field, t a parameter, F=K(t) and f in K[x] a polynomial of
degree d. The polynomial P_n(x,t)= f^n(x) - t in F[x] where f^n is the n-fold
iterate of f, is absolutely irreducible over F; we compute a recursion for its
discriminant. Let L=L(f) be the field obtained by adjoining to F all roots, in
a fixed algebraic closure, of P_n for all n; its Galois group Gal(L/F) is the
iterated monodromy group of f. The iterated extension L/F is finitely ramified
if and only if f is post-critically finite (pcf). We show that, moreover, for
pcf polynomials f, every specialization of L/F at t=t_0 in K is finitely
ramified over K, pointing to the possibility of studying Galois groups with
restricted ramification via tree representations associated to iterated
monodromy groups of pcf polynomials. We discuss the wildness of ramification in
some of these representations, describe prime decomposition in terms of certain
finite graphs, and also give some examples of monogene number fields.Comment: 19 page
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