with the exact value $T_c=0.8076068...$ within an accuracy of the $0.025\%$.

![ (Color online) Finite size scaling of the fourth order cumulant for $q=6$. (a) $V_L$ as a function of temperature for different system sizes. Averages were taken over several samples ranging from $300$ to $400$ for small system sizes down to $50$ and $20$ for $L=128$ and $L=256$. The orange line indicate the analytically predicted location of the minimum in the thermodynamic limit. (b) Pseudo critical temperature $T_c^*$ [*vs.*]{} $1/L^2$. Error bars, estimated from the uncertainty when locating the minimum of $V_L$, are shown only when larger than the symbol size. []{data-label="VL-q6"}](fig3.eps)

Let us emphasize that, as it is well known, it’s very difficult to get good measures of cumulants with a single spin flip MC algorithm. In order to get reliable averages of the cumulant minimum location, one should guarantee a measurement time long enough to let the system overcome the phase separating energy barrier back and forward several times. Moreover, the characteristic activation time to overcome the barrier increases both with $q$ and $L$ (it increases exponentially with $L$). For instance, simulation times of the order $10^7$ for each temperature are needed to obtain a good sampling 