  In this paper we propose a novel method to study critical systems numerically
by a combined collective-mode algorithm and Renormalization Group on the
lattice. This method is an improved version of MCRG in the sense that it has
all the advantages of cluster algorithms. As an application we considered the
2D Ising model and studied wether scale invariance or universality are possible
underlying mechanisms responsible for the approximate "universal fluctuations"
close to a so-called bulk temperature $T^*(L)$. "Universal fluctuations" was
first proposed in [1] and stated that the probability density function of a
global quantity for very dissimilar systems, like a confined turbulent flow and
a 2D magnetic system, properly normalized to the first two moments, becomes
similar to the "universal distribution", originally obtained for the
magnetization in the 2D XY model in the low temperature region. The results for
the critical exponents and the renormalization group flow of the probability
density function are very accurate and show no evidence to support that the
approximate common shape of the PDF should be related to both scale invariance
or universal behavior.
