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

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