10 research outputs found
Ricci curvature on polyhedral surfaces via optimal transportation
The problem of defining correctly geometric objects such as the curvature is
a hard one in discrete geometry. In 2009, Ollivier defined a notion of
curvature applicable to a wide category of measured metric spaces, in
particular to graphs. He named it coarse Ricci curvature because it coincides,
up to some given factor, with the classical Ricci curvature, when the space is
a smooth manifold. Lin, Lu & Yau, Jost & Liu have used and extended this notion
for graphs giving estimates for the curvature and hence the diameter, in terms
of the combinatorics. In this paper, we describe a method for computing the
coarse Ricci curvature and give sharper results, in the specific but crucial
case of polyhedral surfaces
Ricci-flat cubic graphs with girth five
We classify all connected, simple, 3-regular graphs with girth at least 5
that are Ricci-flat. We use the definition of Ricci curvature on graphs given
in Lin-Lu-Yau, Tohoku Math., 2011, which is a variation of Ollivier, J. Funct.
Anal., 2009. A graph is Ricci-flat, if it has vanishing Ricci curvature on all
edges. We show, that the only Ricci-flat cubic graphs with girth at least 5 are
the Petersen graph, the Triplex and the dodecahedral graph. This will correct
the classification in Lin-Lu-Yau, Comm. Anal. Geom., 2014, that misses the
Triplex
Introducing Quantum Ricci Curvature
Motivated by the search for geometric observables in nonperturbative quantum
gravity, we define a notion of coarse-grained Ricci curvature. It is based on a
particular way of extracting the local Ricci curvature of a smooth Riemannian
manifold by comparing the distance between pairs of spheres with that of their
centres. The quantum Ricci curvature is designed for use on non-smooth and
discrete metric spaces, and to satisfy the key criteria of scalability and
computability. We test the prescription on a variety of regular and random
piecewise flat spaces, mostly in two dimensions. This enables us to quantify
its behaviour for short lattices distances and compare its large-scale
behaviour with that of constantly curved model spaces. On the triangulated
spaces considered, the quantum Ricci curvature has good averaging properties
and reproduces classical characteristics on scales large compared to the
discretization scale.Comment: 43 pages, 27 figure
Self-Assembly of Geometric Space from Random Graphs
We present a Euclidean quantum gravity model in which random graphs
dynamically self-assemble into discrete manifold structures. Concretely, we
consider a statistical model driven by a discretisation of the Euclidean
Einstein-Hilbert action; contrary to previous approaches based on simplicial
complexes and Regge calculus our discretisation is based on the Ollivier
curvature, a coarse analogue of the manifold Ricci curvature defined for
generic graphs. The Ollivier curvature is generally difficult to evaluate due
to its definition in terms of optimal transport theory, but we present a new
exact expression for the Ollivier curvature in a wide class of relevant graphs
purely in terms of the numbers of short cycles at an edge. This result should
be of independent intrinsic interest to network theorists. Action minimising
configurations prove to be cubic complexes up to defects; there are indications
that such defects are dynamically suppressed in the macroscopic limit. Closer
examination of a defect free model shows that certain classical configurations
have a geometric interpretation and discretely approximate vacuum solutions to
the Euclidean Einstein-Hilbert action. Working in a configuration space where
the geometric configurations are stable vacua of the theory, we obtain direct
numerical evidence for the existence of a continuous phase transition; this
makes the model a UV completion of Euclidean Einstein gravity. Notably, this
phase transition implies an area-law for the entropy of emerging geometric
space. Certain vacua of the theory can be interpreted as baby universes; we
find that these configurations appear as stable vacua in a mean field
approximation of our model, but are excluded dynamically whenever the action is
exact indicating the dynamical stability of geometric space. The model is
intended as a setting for subsequent studies of emergent time mechanisms.Comment: 26 pages, 9 figures, 2 appendice