getting a well defined probability transfer matrix $\hat W$. The Table \[Example4x4\], reproduced in Appendix \[App1\], summarizes the calculations and main characteristics for various energies.

Next, we shall explore few cycles for larger systems ($8\times 8$ and $16\times 16$). The cycles in these cases may be as long as desired for any practical purpose, so that the equilibrium distribution is solved with enough precision.

In the case of $8\times 8$, for various energies and the finest coarse graining, as a sake of brevity, we omit explicitly the plots of the first eigenvector, ${\bm f}_{\rm eq}$, as well as the eigenvalues, because they are similar to the one of the $16\times 16$ lattice case.

The case of a $16\times 16$ system presents the most accurate equilibrium distributions found in the current research. The fluctuations around the distribution are small, and the eigenvalues seems to form a continuous spectrum (the difference among two consecutive eigenvalues is small). We have also explored a wide range of energies. The rank of the matrices (that is the finest partition) are $K= 122$ for $E= -332$; $K=205$ for $E= -316$; $K= 197$ for $E= -292$; $K=129$ for $E=-168$; and $K= 101$ for $E=-92$. The equilibrium distribution, 