Fig. 1.

TL exponent b for different values of the transition probability λ. The sample exponents computed in R simulations of a two-state multiplicative process with symmetric transition matrix in the two regimes $1\ll t\ll \log R$ (black solid circles, $R={10}^6$ up to time $t=10$) and $t\gg \log R$ (red open squares, $R={10}^4$ up to time $t=400$) are in good agreement with predictions for the asymptotic population (black solid line, Eq. 9) and sample (red dashed line, $b=2$) exponents. In the simulations, the sample exponent b was computed by least-squares fitting of $\log \mathrm{Var}\left[N\left(t\right)\right]$ as a function of $\log \mathbb{E}\left[N\left(t\right)\right]$ for the last 6 (black circles) and 200 (red squares) time steps. In A, which has the plotted theoretical result from ref. 24, $\chi =\left\{r,s\right\}=\left\{2,1/4\right\}$ (b displays a discontinuity); in B, $\chi =\left\{r,s\right\}=\left\{4,1/2\right\}$ (in such a case, b displays no discontinuity). Fig. S1 shows the generalized TL exponent $b_{23}$ in the same simulations.