Fig. 4.

The effective potential at coarse-grained scales. Effective potential for the averaged activity ρ measured in cells of linear size m, in a square lattice of size $N=256\times 256$ (segmentation of the system into boxes schematically illustrated in the insets). The potential is defined for each value of m as $-\log \left[\text{Prob}\left({\rho }_m\right)\right]$, where $\text{Prob}\left({\rho }_m\right)$ is the steady state probability distribution of the activity ρ averaged in boxes of linear size m with (A) $m=1$ and (B) $m=64$. Colors represent different values of a, namely, $a=0.15,0.11,0.07$ and $a=0.521,0.522,0.523,0.524$, respectively (other parameters: $b=-2,c=1,{\sigma }^2=1,D=0.1$). As the coarse-graining scale m is increased, the shape of the effective potential changes, from that typical of discontinuous transitions (for $m=1$) to the one characteristic of continuous ones (at larger coarse-graining scales, e.g., $m=64$). This is tantamount to saying that the renormalized value of b changes sign, from $b<0$ to $b>0$, and that even if the deterministic potential exhibits a discontinuous transition, the renormalized one does not.