that the net polarization drops gradually and the polar transition shifts towards higher $\text{g}$ as we decrease the particles persistence down to $\text{Pe}<1$. The limit $\text{Pe}\to0$ can be approached by taking $v_0\to0$ at fixed $\gamma$ or, conversely, $\gamma\to\infty$ at fixed $v_0$. In the first case, $v_0\to0$, particles generate a Wigner-like crystal, because of the absence of translational noise and the repulsive nature of the interaction potential. Moreover, at the density considered here, the system cannot find a globally oriented state, even in the limit $\text{g}\to\infty$, since the interaction network is not simply connected (a geometrical percolation occurs above $R_{\theta}\approx \sqrt{4.51/\rho\pi}$ [@Dall2002]$\approx3.6\sigma$ here). We shall note that, even at higher densities the system would not be able to sustain a net polarization, since the Mermin-Wagner theorem forbids any 2D static network to globally order. The emergence ofglobal order in models of flocking is purely driven by self-propulsion. In the limit of diverging noise intensity $\gamma\to\infty$, which is the way we approach here the limit $\text{Pe}\to0$, the alignment interaction has to be increasingly large to compete with the noise term eq. \[theta\_dot\] and the motion of the particles becomes basically Brownian. In such a large noise amplitude regime, an agreement with the 