A Quantitative Clustering Approach to Ultrametricity in Spin Glasses

We discuss the problem of ultrametricity in mean field spin glasses by means of a hierarchical clustering algorithm. We complement the clustering approach with quantitative testing: we discuss both in some detail. We show that the elimination of the (in this context accidental) spin flip symmetry plays a crucial role in the analysis, since the symmetry hides the real nature of the data. We are able to use in the analysis disorder averaged quantities. We are able to exhibit a number of features of the low $T$ phase of the mean field theory, and to claim that the full hierarchical structure can be observed without ambiguities only on very large lattice volumes, not currently accessible by numerical simulations.