### 3.6 Evidence for a second-order transition?

The same -measure was used by Hamber in an extension of a previous simulation of the
higher-derivative action (16) [113], for lattice sizes of up to 16^{4}. At the transition point (,
, ), the distributions of edge lengths, volumes and curvatures are smooth
and Gaussian-like, and the average curvature vanishes. The location of from fits (17)
to the average curvature coincide with those from [111], leading to . The data at
do not seem to match this interpretation. This leads to the tentative conclusion that only
for sufficiently large the observed transition is of second order. It is in general “difficult
to entirely exclude the presence of a weak first-order transition, if it has a very small latent
heat”. For , one finds some evidence for a decrease in the fractal dimension as
grows.