We investigate the metric nature of spectral triples in two ways.
Given an oriented Riemannian embedding i:X-\u3eY of codimension 1 we construct a family of unbounded KK-cycles i!(epsilon), each of which represents the shriek class of i in KK-theory. These unbounded KK-cycles are further equipped with connections, allowing for the explicit computation of the products of i! with the spectral triple of Y at the unbounded level. In the limit epsilon to 0 the product of these unbounded KK-cycles with the canonical spectral triple for Y admits an asymptotic expansion. The divergent part of this expansion is known and universal, the constant term in the expansion gives the canonical spectral triple for X. Furthermore, the curvature of these unbounded KK-cycles converges to the square of the mean curvature of X in Y as epsilon goes to 0.
We define a random matrix ensemble for the Dirac operator on the (0,1) fuzzy geometry incorporating both the geometric and fermionic aspects of the spectral action. This yields a unitarily invariant, single-matrix multi-trace model. We generalize Coulomb-gas techniques for finding the spectral density of single-trace models to multi-trace models and apply these to our model of a fermionic fuzzy geometry. The resulting Fredholm integral equation for the spectral density is analyzed numerically and the effect of various parameters on the spectral density is investigated
