7 gular). The first one is non-perturbative in nature, as it provides a cutoff on the number of independent degrees of freedom characterizing the geometry. It also suggests that the system might have some ’granular’ features. The second class of corrections can be used to represent the ordinary quantum fluctuations of the geometry. Given a wave function for these degrees of freedom, we would ask the following simple questions: Notice that when we consider N finite, the notion of topology and shape becomes fuzzy, due to both the fluctuations of the eigenvalues, as well as because of the discreteness of the eigenvalues themselves. First, the eigenvalues have some probability to be in any distribu- tion, except on sets of measure zero. Thus the support of the distribution is not a very useful notion of geometry, but maybe it is enough to describe the topology. However, if one is trying to find sets of measure zero numerically this might not be easy to extract from a simulation in the general case. Secondly, measurements on fluctuating object shapes will have some inherent randomness and there will be errors in measurements. If the particles can be in any spatial distribution, there are no minimal distances to define sizes either. Thus how one chooses to sample the distribution to define a notion of size becomes important as well. In such a situation, we have to change the questions to something that makes sense computationally. We need observables such that in the thermodynamic limit ( N → ∞ ) they recover the classical measurement problem with probability arbitrarily close to one at finite sampling (taking a single measurement on a typical configuration, within some prescribed error bars). Some observables are better suited to this task than others. Moreover, different observables will differ at finite N by 1 /N corrections (see for example the discussion in [36] for fuzzy sphere membranes). So the real question we need to ask is what are the optimal choices of observables that characterize geometric measurements and how fast do they approach the thermodynamic large N limit? As we said above, we will approach this problem by doing numerical simulations. The basic setup for these simulations was described in [21] and [7]. The Monte-Carlo model we 1. What is the topology of the distribution? 2. What are the geometric measurements of the holes of the geometry? 3. How big are the quantum fluctuations?