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In recent times, the theoretical study of the three-dimensionalEdwards-Anderson model has produced several rigorous results on the nature ofthe spin-glass phase. In particular, it has been shown that, as soon as theoverlap distribution is non-trivial, ultrametricity holds. However, thesetheorems are valid only in the thermodynamical limit and are therefore ofuncertain applicability for (perennially off-equilibrium) experimental spinglasses. In addition, their basic assumption of non-triviality is still hotlydebated. This paper intends to show that the predictions stemming fromultrametricity are already well satisfied for the lattice sizes where numericalsimulations are possible (i.e., up to $V = 32^3$ spins) and are, therefore,relevant at experimental scales. To this end we introduce a three-replicacorrelation function, which evinces the ultrametric properties of the systemand is shown to scale in the same way as the overlap correlation function.

Ultrametricity and long-range correlations in the Edwards-Anderson spin glass