By using suitable Monte Carlo methods and finite-size scaling, an investigation was made of the Blume-Capel model on a square lattice. Percolation clusters were constructed by placing nearest-neighbor bonds between vacancies, with a variable bond probability, pb. At the tricritical point, the percolation threshold of these vacancy clusters was located at pbc = 0.706. At this point, the fractal dimension of the vacancy clusters was determined to be Xf = 0.1308 ≈ 21/160, and the exponent which governed the renormalization flow in the pb direction was yp = 0.426 ≈ 17/40. For a bond probability, pb>pbc, the vacancy clusters maintained strong critical correlations, the fractal dimension was Xf = 0.0750 ≈ 3/40 and the leading correction exponent was yp = –0.45 ≈ –19/40. The above values fitted well into the Kac table for the tricritical Ising model. These vacancy clusters had a close analogy with those consisting of Ising spins of the same sign, although the associated quantities, μ, and magnetization, m, were energy-like and magnetic quantities, respectively. However, along the critical line of the Blume-Capel model, the vacancies were more or less uniformly distributed over the whole lattice. In this case, no critical percolation correlations were observed in the vacancy clusters; at least in the physical region pb << 1.

Percolation between Vacancies in the Two-Dimensional Blume-Capel Model. Y.Deng, W.Guo, H.W.J.Blöte: Physical Review E, 2005, 72[1], 016101 (11pp)