1 3 10 4 10 5 10 6 (t) t  0.1  1  1  10 S(k,t) k Q=0.0  0.1  1  1  10 k Q=0.25 t=1.0 10 4 t=2.0 10 4 t=5.0 10 4 t=1.0 10 5 t=2.5 10 5 t=5.5 10 5 S(k,t) -1 k 2 2 Q =0.0 Q =0.25 0 0 t=1000000 MCsteps t=40000 MCsteps FIG. 2: S ( k, t ) at different times for the unconstrained (left) and constrained (right, ( ˆ Q = 0 . 25) cases. Left inset: cor- relation length ξ ( t ) as extracted from Eq. (2). Right inset: S ( k, t ) − 1 vs. k $^{2}$. T = T$_{MC}$ and L = 16. FIG. 1: Fluctuations of the overlap field, δq ( r , t ) = q ( r , t ) − 〈 q ( t ) 〉 for a 2- d slice of the system. Upper panels: uncon- strained system. Lower panels: constrained system ( ˆ Q = 0 . 25). Left panels: t = τ$_{α}$ . Right panels: large times. L = 16. What happens beyond τ$_{α}$ ? The memory of the initial configuration is gradually lost, so that the correlation function S ( k, t ) decays sharply, Fig. 2 (left). What hap- pens to ξ ( t ) is less clear, because the vanishing of S ( k, t ) makes it hard to fit a reliable value of ξ ( t ) through Eq. 2. Although this point is debated [14], our results indicate that ξ ( t ) decreases beyond τ$_{α}$ (left inset of Fig. 2), in line with other studies [13]. However, what is important, and definitely out of question, is that the correlation de- creases for large times, irrespective of its spatial range. Heterogeneities blur as q ( r , t ) becomes zero everywhere, and their size ξ ( t ) becomes somewhat ill-defined. if there is a nonzero surface tension between high and low overlap regions, such a scenario is not what we ex- pect: the surface tension would force different domains to merge, driving the system towards a phase-separated, highly correlated state [15]. Hence, the second hypoth- esis is that the correlation does not decay and that the dynamic correlation length ξ ( t ) grows beyond ξ$_{d}$ , up to an asymptotic value of the order of the system’s size L . The stark difference between these two hypotheses sug- gests that the constrained experiment may clarify the mechanisms of formation of dynamical heterogeneities. Inspection of the overlap field in the constrained case, Fig. 1 (bottom), is quite telling: for large t the system phase separates into high and low overlap regions, form- ing stable dynamical heterogeneities of the order of the system size. From a quantitative point of view the sit- uation is equally clear: in contrast with the free case, the constrained correlation function does not go to zero for large times, but saturates at a finite value, Fig. 2 (right). Hence, even in the late time regime dynamic heterogeneities remain strongly correlated. Let us now make a different experiment. We want to impose a constraint on the dynamics, so that the sys- tem cannot entirely lose memory of its initial configu- ration. This can be implemented by imposing a lower bound on the global overlap, Q ( t ) = 1 /V ∫ d r q ( r , t ). In the unconstrained case Q ( t ) goes asymptotically to zero, as the memory of the initial configuration fades [29]. On the other hand, if we run the dynamics with the con- straint Q ( t ) ≥ ˆ Q things change [30]. Initially the system does not feel the constraint: the global overlap decreases from its t = 0 value, Q = 1, and everything proceeds as described above, including the growth of the hetero- geneities. However, at later times Q ( t ) hits its lower bound ˆ Q and it cannot decrease further. What happens to the dynamical heterogeneities in this case? The study of the correlation length in the constrained case confirms this scenario [31]. In an infinite system undergoing phase separation, or at a critical point, the intercept A in Eq. 2 vanishes while the slope B remains finite, so that ξ grows indefinitely [12]. On the other hand, in a finite system phase separation means that ξ becomes comparable with system size L . As a con- sequence, the finite-size (periodic) real space correlation function G ( r , t ) ceases to be a simple exponential for large r (small k ). This implies that the intercept A in Eq. 2 can go below zero and take small negative values ( O (1 /L $^{2}$)). In a finite-size system it is therefore convenient to com- pare A/B = ξ − 2 vs. L − 2 to check whether or not phase separation occurs. From Fig. 3 (left) we see that in the There are two alternative hypotheses. First, the corre- lation S ( k, t ) and its spatial range ξ ( t ) decay to zero for large t as in the free case. Due to the constraint, however, such endgame cannot happen in the same manner as in the free case, i.e. with q ( r , t ) becoming zero everywhere. Heterogeneities must thus become very small, forming a salt-and-pepper configuration of the field q ( r , t ), so that the total integral of the field stays equal to ˆ Q . Yet,