[]{data-label="fig:density"}](Fig3submission.eps){width="\columnwidth"}

The comparison of the particle-number distribution $p_{{{\tilde{t}}}}(m)$ in the model, varying with the particle density $\rho$, and the empirical share distribution may help us to estimate $\rho$. We perform simulations of the model with $N=508$ sites and different particle densities set to be integers. Particles are uniformly distributed over all sites initially at $\tilde{t}=0$, yielding $p_{\tilde{t}=0}(m) = \delta_{m,\rho}$. As the stochastic transfers of particles are repeated, the particle-number distribution gets broadened as shown in Fig. \[fig:density\] (a). The second moment $\langle A^2 \rangle = M^{-2} \langle m^2\rangle = M^{-2}\sum_m m^2 p_{\tau}(m)$ increases with the rescaled time $\tau$ and saturates in the late-time regime as shown in Fig. \[fig:density\] (b). The empirical value of $\langle A^2\rangle$ increases steadily from $2\times 10^{-5}$ to $4\times 10^{-5}$ for the period 1962-2000 except for the jumps related to oil crises in the middle of the period, which disappear if crude oil (SITC 3330) is excluded \[Fig. \[fig:empirical\] (a)\]. Interestingly, the simulated values of $\langle A^2\rangle$ overlap with the empirical values at $90 \leq \tau\leq 130$ as long as $\rho\gtrsim 10$ \[Fig. \[fig:density\] (b)\]. This time interval $\Delta \tau \simeq 40$ is in quite good agreement with the real time interval, 38 years from 1962 