Figure 2.

Dependence of the logarithm (base 10) of the number of terms, n, to define the partition function versus the number of sites, N, in the lattice for the brute force approach (•) and the approach based on Eq. 7 for finite σ (○) and for σ = 0 (▪). Continuous lines are the best-fits of logn according to the expression logn = a + bN, with: (•) a = 0, b = log2; (○) a = −1.1 ± 0.2, b = 0.29 ± 0.01; (▪) a = −0.36 ± 0.06, b = 0.087 ± 0.002. Note how the method based on Eq. 7 scales as 2^N for finite σ, but involves a number of terms about 13-fold smaller compared with the brute force approach. For σ = 0, the method based on Eq. 7 scales as 1.2^N, thereby involving 1.7^N fewer terms than the brute force approach.