Equilibrium valleys in spin glasses at low temperature

We investigate the three-dimensional Edwards-Anderson spin glass model at low temperature on simple cubic lattices of sizes up to $L=12.$ Our findings show a strong continuity among $Tg0$ physical features and those found previously at $T=0,$ leading to a scenario with emerging mean field like characteristics that are enhanced in the large volume limit. For instance, the picture of space filling sponges seems to survive in the large volume limit at $Tg0,$ while entropic effects play a crucial role in determining the free-energy degeneracy of our finite volume states. All of our analysis is applied to equilibrium configurations obtained by a parallel tempering on $512$ different disorder realizations. First, we consider the spatial properties of the sites where pairs of independent spin configurations differ and we introduce a modified spin overlap distribution which exhibits a nontrivial limit for large L. Second, after removing the ${Z}_{2}$ $(\ifmmode\pm\else\textpm\fi{}1)$ symmetry, we cluster spin configurations into valleys. On average these valleys have free-energy differences of $O(1),$ but a difference in the (extensive) internal energy that grows significantly with L; there is thus a large interplay between energy and entropy fluctuations. We also find that valleys typically differ by spongelike space filling clusters, just as found previously for low-energy system-size excitations above the ground state.