11,590 research outputs found
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
Combinatorics and geometry of finite and infinite squaregraphs
Squaregraphs were originally defined as finite plane graphs in which all
inner faces are quadrilaterals (i.e., 4-cycles) and all inner vertices (i.e.,
the vertices not incident with the outer face) have degrees larger than three.
The planar dual of a finite squaregraph is determined by a triangle-free chord
diagram of the unit disk, which could alternatively be viewed as a
triangle-free line arrangement in the hyperbolic plane. This representation
carries over to infinite plane graphs with finite vertex degrees in which the
balls are finite squaregraphs. Algebraically, finite squaregraphs are median
graphs for which the duals are finite circular split systems. Hence
squaregraphs are at the crosspoint of two dualities, an algebraic and a
geometric one, and thus lend themselves to several combinatorial
interpretations and structural characterizations. With these and the
5-colorability theorem for circle graphs at hand, we prove that every
squaregraph can be isometrically embedded into the Cartesian product of five
trees. This embedding result can also be extended to the infinite case without
reference to an embedding in the plane and without any cardinality restriction
when formulated for median graphs free of cubes and further finite
obstructions. Further, we exhibit a class of squaregraphs that can be embedded
into the product of three trees and we characterize those squaregraphs that are
embeddable into the product of just two trees. Finally, finite squaregraphs
enjoy a number of algorithmic features that do not extend to arbitrary median
graphs. For instance, we show that median-generating sets of finite
squaregraphs can be computed in polynomial time, whereas, not unexpectedly, the
corresponding problem for median graphs turns out to be NP-hard.Comment: 46 pages, 14 figure
Geometric Spanning Cycles in Bichromatic Point Sets
Given a set of points in the plane each colored either red or blue, we find
non-self-intersecting geometric spanning cycles of the red points and of the
blue points such that each edge of the red spanning cycle is crossed at most
three times by the blue spanning cycle and vice-versa
- …