Shape-Driven Nested Markov Tessellations

A new and rather broad class of stationary (i.e. stochastically translation invariant) random tessellations of the $d$-dimensional Euclidean space is introduced, which are called shape-driven nested Markov tessellations. Locally, these tessellations are constructed by means of a spatio-temporal random recursive split dynamics governed by a family of Markovian split kernel, generalizing thereby the -- by now classical -- construction of iteration stable random tessellations. By providing an explicit global construction of the tessellations, it is shown that under suitable assumptions on the split kernels (shape-driven), there exists a unique time-consistent whole-space tessellation-valued Markov process of stationary random tessellations compatible with the given split kernels. Beside the existence and uniqueness result, the typical cell and some aspects of the first-order geometry of these tessellations are in the focus of our discussion

An exact quantification of backreaction in relativistic cosmology

An important open question in cosmology is the degree to which the Friedmann-Lemaitre-Robertson-Walker (FLRW) solutions of Einstein's equations are able to model the large-scale behaviour of the locally inhomogeneous observable universe. We investigate this problem by considering a range of exact n-body solutions of Einstein's constraint equations. These solutions contain discrete masses, and so allow arbitrarily large density contrasts to be modelled. We restrict our study to regularly arranged distributions of masses in topological 3-spheres. This has the benefit of allowing straightforward comparisons to be made with FLRW solutions, as both spacetimes admit a discrete group of symmetries. It also provides a time-symmetric hypersurface at the moment of maximum expansion that allows the constraint equations to be solved exactly. We find that when all the mass in the universe is condensed into a small number of objects (<10) then the amount of backreaction in dust models can be large, with O(1) deviations from the predictions of the corresponding FLRW solutions. When the number of masses is large (>100), however, then our measures of backreaction become small (<1%). This result does not rely on any averaging procedures, which are notoriously hard to define uniquely in general relativity, and so provides (to the best of our knowledge) the first exact and unambiguous demonstration of backreaction in general relativistic cosmological modelling. Discrete models such as these can therefore be used as laboratories to test ideas about backreaction that could be applied in more complicated and realistic settings.Comment: 13 pages, 9 figures. Corrections made to Tables IV and

Honeycomb tessellations and canonical bases for permutohedral blades

This paper studies two families of piecewise constant functions which are determined by the $(n-2)$-skeleta of collections of honeycomb tessellations of $\mathbb{R}^{n-1}$ with standard permutohedra. The union of the codimension $1$ cones obtained by extending the facets which are incident to a vertex of such a tessellation is called a blade. We prove ring-theoretically that such a honeycomb, with 1-skeleton built from a cyclic sequence of segments in the root directions $e_i-e_{i+1}$, decomposes locally as a Minkowski sum of isometrically embedded components of hexagonal honeycombs: tripods and one-dimensional subspaces. For each triangulation of a cyclically oriented polygon there exists such a factorization. This consequently gives resolution to an issue proposed and developed by A. Ocneanu, to find a structure theory for an object he discovered during his investigations into higher Lie theories: permutohedral blades. We introduce a certain canonical basis for a vector space spanned by piecewise constant functions of blades which is compatible with various quotient spaces appearing in algebra, topology and scattering amplitudes. Various connections to scattering amplitudes are discussed, giving new geometric interpretations for certain combinatorial identities for one-loop Parke-Taylor factors. We give a closed formula for the graded dimension of the canonical blade basis. We conjecture that the coefficients of the generating function numerators for the diagonals are symmetric and unimodal.Comment: Added references; new section on configuration space