$m$ differently depending on whether $\rho$ is larger than $\rho_c$ or not. Consequently, for $\rho\lesssim \rho_c$, $p_\infty(m)$ has an exponential-decay factor $$p_\infty(m) \simeq {1\over \zeta(2) m^2} e^{-\mu\, m}
\label{eq:pmasym1}$$ with $\mu = e^{-\zeta(2) \rho}>0$. In contrast, for $\rho\gg \rho_c$, it has a bump, in addition to the power-law decay form small $m$, as $$p_\infty\left(m = M - {N\over \zeta(2)} \log {2N\over\zeta(2)} + {N\over \zeta(2)} y\right) \simeq 
{\zeta(2)\over N^2}  {1\over 2\sqrt{\pi e}} e^{ -{1\over 4} y^2 }
\label{eq:pmasym2}$$ for $y$ finite, indicating the emergence of a pseudo-condensate [@Evans:2006uq]: Most particles gather at a single site. Given $N=508$, the estimated particle density $\rho\gtrsim 10$ is above the critical density $\rho_c\simeq 3.5$ \[Appendix  \[sec:pm\]\] suggesting that the kinetics of product share will enter the pseudo-condensate phase in the long-time limit: The share distribution is expected to get broadened and end up with a power-law decay plus a bump near $A=1$ as an example in Fig. \[fig:density\] (a) for $\rho=50$.

The time scale for such condensation phenomena can be predicted by our simulation results. $\langle A^2\rangle$ saturates around $\tau\simeq 10^3$ for $\rho\gtrsim 10$ \[Fig. \[fig:density\] (b)\], which suggest that the stationary-state distribution displaying a bump will appear in $10^3$ years.

The predictions of the 