Monte-Carlo sampling of triangulations of near mean-action zero.

<p>We plot the distribution of mean actions at for 2700 sampled triangulations of the 3-sphere . Samples were obtained from a Metropolis-Hastings algorithm using Pachner moves and a quadratic objective function targeting and with and . Waiting times were chosen so that accepted moves per tetrahedra occurred between successive samples. Observed means were with standard deviation and with standard deviation . Note that and are given in Planck units, and respectively.</p

Entropy remains a decreasing function of mean-action as the number of tetrahedra grows.

<p>We plot the change in spacetime entropy, in bits, due to each minimal increase in mean-action for the 3-sphere near , versus mean number of tetrahedra . Values were inferred from the bias seen in Monte-Carlo samples of triangulations near . See <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0080826#pone-0080826-g001" target="_blank">Figure 1</a>. All data points except the last two were computed from 2700 samples. At the two largest values, we used 2394 and 1108 samples respectively. Error bars indicate 95% confidence intervals.</p

Meaning of Commonly Used Symbols.

<p>In this table we list some of the commonly used symbols in this paper and their meanings.</p

Entropy versus mean-action from triangulation census data.

<p>We plot spacetime entropy in bits for the three-sphere at various numbers of tetrahedra , versus mean action at . Data come from a complete census <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0080826#pone.0080826-Burton1" target="_blank">[51]</a>, <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0080826#pone.0080826-Burton2" target="_blank">[52]</a> of the million triangulations of with at most 9 tetrahedra. Note that and are given in Planck units, and respectively.</p