Exact cluster size distributions and mean cluster sizes for the q-state bond-correlated percolation model

The author shows that the q-state bond-correlated percolation model (QBCPM), which is the percolation representation of the q-state Potts model (QPM), on the lattice without closed loops is equivalent to the bond random percolation model (BRPM) on the same lattice. Using such results and exact results for the BRPM on the linear and Bethe lattices, the author obtains exact cluster size distributions and the mean cluster size S for the QBCPM on the linear and Bethe lattices. The mean cluster sizes obtained from this method are the same as those obtained by more tedious exact calculations. Near the critical point, the average number of m site clusters per site, nm, for the QBCPM on the linear and Bethe lattices may be written in the scaling form for large values of m, which is the geometrical basis for the scaling laws of critical exponents.

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