$\phi(\eta)= \int_{-\infty}^0 {dx \over 2\pi i} e^{-\eta ( x \log (x-i\epsilon) -  x)} + \int_0^{-\infty} {dx \over 2\pi i} e^{-\eta ( x \log (x+i\epsilon) -  x)}=\int_0^{\infty} {dx \over \pi} e^{\eta ( x - x \log x)} \sin (\eta \pi x)$, which leads us to $$\begin{aligned}
\phi(\eta) &= \int_0^{\infty} {dq \over \eta \pi} e^{ ( q - q \log q + q \log \eta)} \sin (\pi q) \nonumber\\
 &\simeq \int_0^{\infty} {dq \over \eta} q\, e^{  q \log \eta}   \simeq {1\over \eta |\log \eta|^2}.
\label{eq:phi2}\end{aligned}$$

![The particle-number distribution $p_\infty(m)$ in the urn model with $b=2$. (a) $p_\infty(m)$ from simulations decays exponentially for $\rho=2$ while it exhibits a bump for $\rho=10$ at different locations depending on the system size $N$. The dashed line indicates the power-law decay $f_2(m)$ in Eq. (\[eq:pm\]). (b) Data collapse of $[m^2 \zeta(2) p_\infty(m)]^{\zeta(2)/N\mu}$ versus $\zeta(2) m/N$ for different $N$’s and $\rho=2$ (fluid phase). The dashed line represents the theoretical prediction $y=e^{1-e^x}$ in Eq. (\[eq:fluid\]). (c) Data collapse of bumps, $[m^2 \zeta(2) p_\infty(m)]/[\log (\zeta(2)/(N\mu))]^2$ versus $\zeta(2)m/N-\log(\zeta(2)/(N\mu))$ for different $N$’s and $\rho=10$ (condensate phase). The dashed line is $y = {1\over \sqrt{2\pi}} e^{x/2 - e^x}$ from Eq. (\[eq:condensate\]). []{data-label="fig:stationary"}](Fig5submission.eps){width="\columnwidth"}

Different behaviors of $\phi(\eta)$ for $\eta \gtrsim 1$ and $\eta\ll 1$ 