The behaviors of $e$ and $m$ as functions of $T$ for different values of $q$ are shown in Fig.\[fig3\]. From these calculations we obtain the values of the energy ($e_d$ and $e_o$) and magnetization jump $\Delta m$ at the exact transition temperature (see Section \[model\]). Results are compared with exact values in table \[valores-medidos-vs-exactos\]

![(Color online) Equilibrium energy per spin $e$ and magnetization $m$ (inset) versus temperature for $q=9,12,15, 96$. Exact values at the transition point from equations (\[Ujump\]), (\[Uadd\]) and (\[magne-exact\]) are marked as crosses. Data comes from averages over $10$ samples of linear system size $L=2048$. Error bars are smaller than the symbol size.[]{data-label="fig3"}](fig2.eps)

\[valores-medidos-vs-exactos\]

  ----- ------------- ------------ ------------- ------------ ------------- ------------
   $q$                                                                      
            exact      calculated      exact      calculated      exact      calculated
    6    1.508980...    1.51(2)     1.307516...    1.306(1)    0.677083...    0.674(2)
    9    1.633167...   1.6332(5)    1.033499...   1.0334(5)    0.834019...   0.8338(4)
   15    1.765905...   1.7659(2)    0.750492...   0.7509(4)    0.916693...   0.9167(3)
   96    1.960306...   1.96030(3)   0.243817...   0.24382(4)   0.989247...   0.98924(2)
  ----- ------------- ------------ ------------- ------------ ------------- ------------

  : Comparison between calculated and known exact values of $e_o$, $e_d$, and $\Delta m$ at the transition for different values of $q$. Results were obtained from averages over $10$ samples of linear system size $L=2048$ and equilibration and measurements times of at least $5\times10^5$ 