MCS each one.

We can see a very good agreement between data and exact results. It’s worth noting that the data from table \[valores-medidos-vs-exactos\] is not the result of extrapolations of some finite size analysis, but the values from curves in Fig.\[fig3\] at the transition itself. Since we measure one point each $\Delta T$ in temperature, cooling and heating procedures won’t necessary lead to a point measured exactly at $T_c$. So, we have to interpolate points close to $T_c$ to deduce the corresponding values of $e_o$, $e_d$ and $m_o$ at $T_c$. The differences obtained from interpolations using points separated by $\Delta T$ and points separated by $2\times \Delta T$ determine the estimated errors.

We also calculate the fourth order cumulant of the energy [@Ja1993; @Bi1997]

$$V_L = 1- \frac{\langle H^4 \rangle}{3 \langle H^2 \rangle^2}$$

as a function of the temperature for $q=6$ and different system sizes. As it is well known, $V_L$ is almost constant far away from the transition temperature and exhibits a minimum at a pseudo critical temperature $$T_c^*(L)  = T_c + \frac{T_c^2 \ln(q e_o^2 / e_d^2)}{e_d - e_o} \frac{1}{L^d}$$

In Fig.\[VL-q6\]b we show $T_c^*(L)$ vs. $1/L^2$ for $q=6$. The extrapolated value of $T_c^*(L)$ for $L\to\infty$, $0.8078\pm0.0002$ agrees 