to 2000. For $\rho\lesssim 10$, the overlap period varies with $\rho$ or does not even exist e.g., for $\rho=2$ or $3$. Based on these results, we estimate the particle density to be $\rho\gtrsim 10$.

Share distribution and condensation
-----------------------------------

For all $\rho\gtrsim 10$, the simulated share distributions, $P_\tau(A) =  p_\tau(m=A\, M) M$ in the period $90\leq \tau\leq 130$ match excellently the empirical share distributions as shown in Fig. \[fig:density\] (c). The simulated $P_\tau(A)$ with $\rho=2$ decays faster, failing to fit into the empirical distributions even in the long-time limit.

Given such excellent agreement between the empirical data and the simulation results in the period $90\leq \tau\leq 130$, one may wonder what will happen in the late-time regime in the simulations. It can hint at the future of international trade. While the broadening of the simulated distribution $P_\tau(A)$ in the early-time regime does not depend on the particle density, $P_\tau(A)$ in the late-time regime varies significantly with $\rho$. The analytic solution for $p_\infty(m)$ in the stationary state, derived in Appendix \[sec:pm\], shows that similarly to $b>2$, a critical density $\rho_c\sim \log N$ exists for $b=2$ and $N$ large but finite such that $Z_{N-1,M-m}/Z_{N,M}$ in Eq. (\[eq:pm\]) behaves as a function of 