I am searching for an elementary proof of the first order approximation of the current of particles in ASEP (asymmetric simple exclusion process) and TASEP (totally asymetric). To avoid technical details unuseful for the intuition, I will focus on the current at the site $0$.

A simple case : step initial condition

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 almost 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. $\eta_0(x)=\mathbf{1}_{x>0}$).

One can explain this result using the density (renormalized) profile of particles in the window $[-\gamma t, \gamma t]$ (or $[-\gamma , \gamma ]$ after renormalization) which is actually a stronger result, but more natural. The fact is that this density profile is linear, hence a number of particles on the left of $0$ like $t\gamma/4$.

Step Bernoulli initial condition

When the initial condition is step Bernoulli, i.e., $\eta_0(x)=\mathbf{1}_{x>0}Ber_x(\rho)$, where $Ber_x(\rho)$ 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 proof of the fluctuations, the phase transition appears clearly in the calculations (when approximating the fredholm determinant for $t\rightarrow +\infty$), but I would like to find a simple physical interpretation ( I mean a simple and/or direct proof) explaining why the approximation does not depend on $\rho$ as soon as $\rho >1/2$.