Stochastic and algorithmic causal sets for de sitter spacetime

Bolognesi T.

Causal sets  Discrete spacetime  De Sitter spacetime  Models of Computation  Computer Science 

Causal sets ('causets') are directed, acyclic graphs used as discrete models of spacetime. A well known technique for obtaining causets, called 'sprinkling', is based on uniformly distributing points on a continuous spacetime, and letting them inherit the causality relation of the latter, as represented by lightcones. In 2012 Krioukov and others have shown that sprinkling on a de Sitter spacetime yields causets which, surprisingly, share important features with complex networks arising in a variety of other fields, e.g. the Internet; one of these features is the power law distribution of node degrees. How frequent are stochastic or deterministic causets with a power law distribution of node degrees? In this paper we show that causets exhibiting this feature can also be obtained directly, i.e. without assuming an initial manifold, by a simple stochastic process based on generating pairs of random integers in a linearly growing range. Then we address the problem of obtaining similar causets by deterministic, algorithmic techniques, and show that the so called fractal sequence -- a structure we have often encountered in simulations of deterministic computational universes -- can indeed be used for obtaining a first, elementary form of algorithmic 'de Sitter causet'. An interactive Mathematica demonstration has been implemented for illustrating the exponential space growth rate of de Sitter spacetime. Furthermore, Mathematica has been used for creating sprinklings and causets in de Sitter spacetime, for implementing the described, alternative procedures for causet construction (both stochastic and deterministic), and for analyzing causet node degree distributions.

Source: UGM 2013 - Mathematica Italia User Group Meeting 2013, pp. 8–22, Bologna, Italy, 29-30-31 Maggio 2013