Influence of the measure

3.5 Influence of the measure

For the pure Einstein action, (12

) with $λ = 0$ , some differences between the $M1 = dl∕l$ - and the $M2 = ldl$ -measure were investigated on a 4⁴-lattice by Beirl et al. [29 , 30 ]. A constant-volume constraint was used for simulations with the $M1$ -measure, and a cutoff $l2 < const$ for $M2$ . A study of the $k$ -dependence of bulk geometric quantities agreed with previous simulations, wherever applicable. For small $k$ , both measures lead essentially to identical results. For the $M1$ -measure only, one finds that $⟨R ⟩∕⟨V ⟩$ and $⟨A ⟩$ exhibit a jump at some $k0$ , due to the formation of spikes (isolated long link lengths, with the areas staying small). From this, and the study of edge length distributions, one concludes that the DeWitt-type measure $M2$ is generally better behaved.