  It has been considered that the effective bending rigidity of fluid membranes
should be reduced by thermal undulations. However, recent thorough
investigation by Pinnow and Helfrich revealed significance of measure factors
for the partition sum. Accepting the local curvature as a statistical measure,
they found that fluid membranes are stiffened macroscopically. In order to
examine this remarkable idea, we performed extensive ab initio simulations for
a fluid membrane. We set up transfer matrix which is diagonalized by means of
the density-matrix renormalization group. Our method has an advantage in that
it allows us to survey various statistical measures. As a consequence, we found
that the effective bending rigidity flows toward strong coupling under the
choice of local curvature as a statistical measure. On the contrary, for other
measures such as normal displacement and tilt angle, we found clear tendency
toward softening.
