7,951 research outputs found
-Colored Graphs - a Review of Sundry Properties
We review the combinatorial, topological, algebraic and metric properties
supported by -colored graphs, with a focus on those that are pertinent
to the study of tensor model theories. We show how to extract a limiting
continuum metric space from this set of graphs and detail properties of this
limit through the calculation of exponents at criticality
Capturing the phase diagram of (2+1)-dimensional CDT using a balls-in-boxes model
We study the phase diagram of a one-dimensional balls-in-boxes (BIB) model
that has been proposed as an effective model for the spatial-volume dynamics of
(2+1)-dimensional causal dynamical triangulations (CDT). The latter is a
statistical model of random geometries and a candidate for a nonperturbative
formulation of quantum gravity, and it is known to have an interesting phase
diagram, in particular including a phase of extended geometry with classical
properties. Our results corroborate a previous analysis suggesting that a
particular type of potential is needed in the BIB model in order to reproduce
the droplet condensation typical of the extended phase of CDT. Since such a
potential can be obtained by a minisuperspace reduction of a (2+1)-dimensional
gravity theory of the Ho\v{r}ava-Lifshitz type, our result strengthens the link
between CDT and Ho\v{r}ava-Lifshitz gravity.Comment: 21 pages, 7 figure
Tensor models and embedded Riemann surfaces
Tensor models and, more generally, group field theories are candidates for
higher-dimensional quantum gravity, just as matrix models are in the 2d
setting. With the recent advent of a 1/N-expansion for coloured tensor models,
more focus has been given to the study of the topological aspects of their
Feynman graphs. Crucial to the aforementioned analysis were certain subgraphs
known as bubbles and jackets. We demonstrate in the 3d case that these graphs
are generated by matrix models embedded inside the tensor theory. Moreover, we
show that the jacket graphs represent (Heegaard) splitting surfaces for the
triangulation dual to the Feynman graph. With this in hand, we are able to
re-express the Boulatov model as a quantum field theory on these Riemann
surfaces.Comment: 9 pages, 7 fi
A Perspective on Unique Information: Directionality, Intuitions, and Secret Key Agreement
Recently, the partial information decomposition emerged as a promising
framework for identifying the meaningful components of the information
contained in a joint distribution. Its adoption and practical application,
however, have been stymied by the lack of a generally-accepted method of
quantifying its components. Here, we briefly discuss the bivariate (two-source)
partial information decomposition and two implicitly directional
interpretations used to intuitively motivate alternative component definitions.
Drawing parallels with secret key agreement rates from information-theoretic
cryptography, we demonstrate that these intuitions are mutually incompatible
and suggest that this underlies the persistence of competing definitions and
interpretations. Having highlighted this hitherto unacknowledged issue, we
outline several possible solutions.Comment: 5 pages, 3 tables;
http://csc.ucdavis.edu/~cmg/compmech/pubs/pid_intuition.ht
Unique Information via Dependency Constraints
The partial information decomposition (PID) is perhaps the leading proposal
for resolving information shared between a set of sources and a target into
redundant, synergistic, and unique constituents. Unfortunately, the PID
framework has been hindered by a lack of a generally agreed-upon, multivariate
method of quantifying the constituents. Here, we take a step toward rectifying
this by developing a decomposition based on a new method that quantifies unique
information. We first develop a broadly applicable method---the dependency
decomposition---that delineates how statistical dependencies influence the
structure of a joint distribution. The dependency decomposition then allows us
to define a measure of the information about a target that can be uniquely
attributed to a particular source as the least amount which the source-target
statistical dependency can influence the information shared between the sources
and the target. The result is the first measure that satisfies the core axioms
of the PID framework while not satisfying the Blackwell relation, which depends
on a particular interpretation of how the variables are related. This makes a
key step forward to a practical PID.Comment: 15 pages, 7 figures, 2 tables, 3 appendices;
http://csc.ucdavis.edu/~cmg/compmech/pubs/idep.ht
Unique Information and Secret Key Agreement
The partial information decomposition (PID) is a promising framework for
decomposing a joint random variable into the amount of influence each source
variable Xi has on a target variable Y, relative to the other sources. For two
sources, influence breaks down into the information that both X0 and X1
redundantly share with Y, what X0 uniquely shares with Y, what X1 uniquely
shares with Y, and finally what X0 and X1 synergistically share with Y.
Unfortunately, considerable disagreement has arisen as to how these four
components should be quantified. Drawing from cryptography, we consider the
secret key agreement rate as an operational method of quantifying unique
informations. Secret key agreement rate comes in several forms, depending upon
which parties are permitted to communicate. We demonstrate that three of these
four forms are inconsistent with the PID. The remaining form implies certain
interpretations as to the PID's meaning---interpretations not present in PID's
definition but that, we argue, need to be explicit. These reveal an
inconsistency between third-order connected information, two-way secret key
agreement rate, and synergy. Similar difficulties arise with a popular PID
measure in light the results here as well as from a maximum entropy viewpoint.
We close by reviewing the challenges facing the PID.Comment: 9 pages, 3 figures, 4 tables;
http://csc.ucdavis.edu/~cmg/compmech/pubs/pid_skar.htm. arXiv admin note:
text overlap with arXiv:1808.0860
Group field theories for all loop quantum gravity
Group field theories represent a 2nd quantized reformulation of the loop
quantum gravity state space and a completion of the spin foam formalism. States
of the canonical theory, in the traditional continuum setting, have support on
graphs of arbitrary valence. On the other hand, group field theories have
usually been defined in a simplicial context, thus dealing with a restricted
set of graphs. In this paper, we generalize the combinatorics of group field
theories to cover all the loop quantum gravity state space. As an explicit
example, we describe the GFT formulation of the KKL spin foam model, as well as
a particular modified version. We show that the use of tensor model tools
allows for the most effective construction. In order to clarify the
mathematical basis of our construction and of the formalisms with which we
deal, we also give an exhaustive description of the combinatorial structures
entering spin foam models and group field theories, both at the level of the
boundary states and of the quantum amplitudes.Comment: version published in New Journal of Physic
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