content_image/337401.jpg content_image/337402.jpg content_image/337403.jpg content_image/337404.jpg content_image/337405.jpg content_image/337406.jpg
therefore 

for the Potts model, and 

for the site percolation. 

<img src='content_image/337401.jpg'>

<img src='content_image/337402.jpg'>

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 

<img src='content_image/337403.jpg'>

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 

<img src='content_image/337404.jpg'>

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 

<img src='content_image/337405.jpg'>

<img src='content_image/337406.jpg'>

