therefore for the Potts model, and for the site percolation. The partitions can again be coded by means of a sequence of integers. A detailed de- scription of the coding algorithm can be found in [18]. The magnetic sectors of the transfer matrix now have the dimensions d ( m ) L × d ( m ) L and d S ( m ) L × d S ( m ) L for the Potts model and site percolation, respectively, and are much larger than those of the non-magnetic sectors. The magnetic sector of the transfer matrix can also be converted into a product of sparse matrices in the same way as in the case of the non-magnetic sector. The inverse magnetic correlation length is given by where λ$_{0}$ and λ 0 are the leading eigenvalues of the transfer matrix in the non-magnetic and magnetic sector respectively, ζ is a geometrical factor which is the ratio between the unit of L and the thickness of a layer added by the transfer matrix. The magnetic scaled gap then follows. According to finite-size scaling theory [25] and Cardy’s conformal mapping [26], X$_{h}$ ( v, L ) can be expanded as where X$_{h}$ is the magnetic scaling dimension, t is the deviation from the critical point, and u the irrelevant field. Here, y$_{t}$ is the thermal renormalization exponent, y$_{u}$ the leading irrelevant renormalization exponent, and a and b are unknown constants. We substitute (24) into the finite-size scaling equation connecting lattices of sizes L and $^{L}$− 1, and denote the solution of (25) by v$_{c}$ ( L ), which has the expansion 13 d ( m ) L = 2 $^{L}$− 1 ∑ m =0 h$_{m}$ (21) d S ( m ) L = 2 $^{L}$− 1 ∑ m =0 h S m (22) 1 ξ$_{h}$ ( v, L ) = ζ ln ( λ$_{0}$ λ ′ $_{0}$) , (23) X$_{h}$ ( v, L ) = X$_{h}$ + atL y t + buL y u + ... , (24) X$_{h}$ ( v, L ) = X$_{h}$ ( $^{v, L}$− 1) , (25) v$_{c}$ ( L ) = v$_{c}$ + a $^{′}$uL y u − y t $^{+}$· · · , (26)