Fig. 2.

Time evolution of the sample TL exponent. The sample exponent {computed as the slope of the curve $\log \mathbb{E}\left[N{\left(t\right)}^2\right]$ vs. $\log \mathbb{E}\left[N\left(t\right)\right]$} crosses over from the approximate population exponent (Eq. 9, dashed upper horizontal line) at small times to $b\simeq 2$ (dotted lower horizontal line) at larger times. The number of simulations $R={10}^n$ increases exponentially from ${10}^2$ (blue dashed lines) to ${10}^6$ (red solid lines), whereas the crossover time increases approximately linearly. Here, $\chi =\left\{r,s\right\}=\left\{2,1/4\right\}$ and the transition probability (with $\mathrm{\Pi }$ symmetric) is $\lambda =0.55$. (A) Theoretical prediction computed via Eq. 15. (B) Simulation results. Curves are averaged over ${10}^8/R$ simulations (except for the blue curve, averaged over ${10}^5$ simulations). Mismatches between A and B are due to the necessity to have t and R not too large to keep simulations feasible, whereas Eqs. 8 and 13 hold true asymptotically in t.