Consider a dynamical many-body system with a random initial state subsequently evolving through stochastic dynamics. What is the relative importance of the initial state (nature) vs. the realization of the stochastic dynamics (nurture) in predicting
the final state? We examined this question for the two-dimensional Ising ferromagnet following an initial deep quench from $T=infty$ to $T=0$. We performed Monte Carlo studies on the overlap between identical twins raised in independent dynamical environments, up to size $L=500$. Our results suggest an overlap decaying with time as $t^{-theta_h}$ with $theta_h = 0.22 pm 0.02$; the same exponent holds for a quench to low but nonzero temperature. This heritability exponent may equal the persistence exponent for the 2D Ising ferromagnet, but the two differ more generally.
In Phys. Rev. Lett. 110, 219701 (2013) [arXiv:1211.0843] Billoire et al. criticize the conclusions of our Letter [Phys. Rev. Lett. 109, 177204 (2012), arxiv:1206.0783]. They argue that considering the Edwards-Anderson and Sherrington-Kirkpatrick mode
ls at the same temperature is inappropriate and propose an interpretation based on the replica symmetry breaking theory. Here we show that the theory presented in the Comment does not explain our data on the Edwards-Anderson spin glass and we stand by our assertion that the low-temperature behavior of the Edwards-Anderson spin glass model does not appear to be mean-field like.
Magnetic ordering at low temperature for Ising ferromagnets manifests itself within the associated Fortuin-Kasteleyn (FK) random cluster representation as the occurrence of a single positive density percolating network. In this paper we investigate t
he percolation signature for Ising spin glass ordering -- both in short-range (EA) and infinite-range (SK) models -- within a two-replica FK representation and also within the different Chayes-Machta-Redner two-replica graphical representation. Based on numerical studies of the $pm J$ EA model in three dimensions and on rigorous results for the SK model, we conclude that the spin glass transition corresponds to the appearance of {it two} percolating clusters of {it unequal} densities.
