the origin of the empirical features of product share presented in Sec. \[sec:empirical\].

![Particle transfer and phase diagram of the urn model. (a) Stochastic transfer of a particle (unit of share) for $N=5$ sites (products) with rate proportional to the square of the number of particles at source and destination sites as $m_p^2 m_q^2$. (b) Phase diagram in the ($b,\rho$) plane with particle density $\rho=M/N$ and the scaling exponent $b$ of Eq. (\[eq:u\]). The boundary between fluid and condensate phase is given by Eq. (\[eq:rhoc\]). []{data-label="fig:model"}](Fig2submission.eps){width="\columnwidth"}

As the first step of modeling the kinetics of product share, we discretize the product share by introducing a unit share $a$, representing the amount available for a single strategic change. Then we find the continuous share $A_p$ discretized to $m_p = A_p/a$ particles, resulting in a total of $M=\sum_{p=1}^N m_p = 1/a$ particles distributed over $N$ urns or sites (products). Transferring a particle from a site to another represents the flow of capital between them caused by the change of investment strategy of microscopic agents on a short time scale. We assume that those [*share*]{} particles hop from a site to another stochastically and independently. Such a particle-hopping model can be classified as 