Figure 5.

Expected values of $p_{n,t;m}$. In (a), we consider the evolution of ${\mu }_{t;m}$ for $m=8$. The solid black curve is the exact evolution predicted by $p_{n,t;m}$, Equation (45), the blue dotted curve depicts the linear behavior predicted by Equation (46), and the red dashed line coincides with the origin. In (b), we show the bounded growth ${\sigma }_{t;m}^2$, for $m=8$. Again, the solid black curve represents the exact Formula (48), the blue dotted curve corresponds to approximate expression (50), while the red dashed line stems from Equation (49). In both cases, $N=16$, and the solid circles were obtained from 100,000 numerical simulations of the process.