I am searching for a "physical" interpretation of the first order approximation of the current of particles in ASEP (asymmetric simple exclusion process) and TASEP (totally asymetric).

It has been known for a long time (see for exemple theorem 5.12 of the section devoted to exclusion processes in "Interacting particle systems", Liggett) that the current of particles at site $0$ in the ASEP model converges ammost surely to $t\gamma/4$ when $t$ tends to infinity, where $\gamma=q-p$, when the initial condition is the so called step initial condition. (i.e. $\eta_0(x)=\mathbf{1}_{x>0}$).

When the initial condition is step Bernoulli, i.e., $\eta_0(x)=\mathbf{1}_{x>0}Ber(\rho)$, where $Ber(\rho)$ is a Bernoulli random variable with parameter $\rho$, the approximation for the current stays the same for $\rho\geq 1/2$, and the fluctuations follow tracy-widom law on a $t^{1/3}$ scale. For $\rho< 1/2$, the current in $0$ is like $t\gamma\rho(1-rho)$ (see for example Tracy-Widom, "On ASEP with Step Bernoulli Initial Condition"), and the fluctuations are Gaussian.

In the calculations, the phase transition appears clearly, but I would like to find a simple physical interpretation explaining why the approximation does not depend on $\rho$ as soon as $\rho >1/2$.