16 research outputs found
Everpresent Lambda - II: Structural Stability
Ideas from causal set theory lead to a fluctuating, time dependent
cosmological-constant of the right order of magnitude to match currently quoted
"dark energy" values. Although such a term was predicted some time ago, a more
detailed analysis of the resulting class of phenomenological models was begun
only recently (based on numerical simulation of the cosmological equations with
such a fluctuating term). In this paper we continue the investigation by
studying the sensitivity of the scheme to some of the ad hoc choices made in
setting it up.Comment: 15 pages, 6 figures, Thoroughly rewritte
Evidence for a continuum limit in causal set dynamics
We find evidence for a continuum limit of a particular causal set dynamics
which depends on only a single ``coupling constant'' and is easy to
simulate on a computer. The model in question is a stochastic process that can
also be interpreted as 1-dimensional directed percolation, or in terms of
random graphs.Comment: 24 pages, 19 figures, LaTeX, adjusted terminolog
A Classical Sequential Growth Dynamics for Causal Sets
Starting from certain causality conditions and a discrete form of general
covariance, we derive a very general family of classically stochastic,
sequential growth dynamics for causal sets. The resulting theories provide a
relatively accessible ``half way house'' to full quantum gravity that possibly
contains the latter's classical limit (general relativity). Because they can be
expressed in terms of state models for an assembly of Ising spins living on the
relations of the causal set, these theories also illustrate how
non-gravitational matter can arise dynamically from the causal set without
having to be built in at the fundamental level. Additionally, our results bring
into focus some interpretive issues of importance for causal set dynamics, and
for quantum gravity more generally.Comment: 28 pages, 9 figures, LaTeX, added references and a footnote, minor
correction
Towards a Definition of Locality in a Manifoldlike Causal Set
It is a common misconception that spacetime discreteness necessarily implies
a violation of local Lorentz invariance. In fact, in the causal set approach to
quantum gravity, Lorentz invariance follows from the specific implementation of
the discreteness hypothesis. However, this comes at the cost of locality. In
particular, it is difficult to define a "local" region in a manifoldlike causal
set, i.e., one that corresponds to an approximately flat spacetime region.
Following up on suggestions from previous work, we bridge this lacuna by
proposing a definition of locality based on the abundance of m-element
order-intervals as a function of m in a causal set. We obtain analytic
expressions for the expectation value of this function for an ensemble of
causal set that faithfully embeds into an Alexandrov interval in d-dimensional
Minkowski spacetime and use it to define local regions in a manifoldlike causal
set. We use this to argue that evidence of local regions is a necessary
condition for manifoldlikeness in a causal set. This in addition provides a new
continuum dimension estimator. We perform extensive simulations which support
our claims.Comment: 35 pages, 17 figure
Probabilistic and analytical properties of the last passage percolation constant in a weighted random directed graph
To each edge (i,j), i<j of the complete directed graph on the integers we
assign unit weight with probability p or weight x with probability 1-p,
independently from edge to edge, and give to each path weight equal to the sum
of its edge weights. If W^x_{0,n} is the maximum weight of all paths from 0 to
n then W^x_{0,n}/n \to C_p(x),as n\to\infty, almost surely, where C_p(x) is
positive and deterministic. We study C_p(x) as a function of x, for fixed 0<p<1
and show that it is a strictly increasing convex function that is not
differentiable if and only if x is a nonpositive rational or a positive integer
except 1 or the reciprocal of it. We allow x to be any real number, even
negative, or, possibly, -\infty. The case x=-\infty corresponds to the
well-studied directed version of the Erd"os-R'enyi random graph (known as
Barak-Erd"os graph) for which C_p(-\infty) = lim_{x\to -\infty} C_p(x) has been
studied as a function of p in a number of papers.Comment: 24 pages, 6 figure