Percolation in two-dimensional lattices. II. The extent of universality

By making use of finite-size scaling and Monte Carlo simulations, we study the so-called ‘‘universality’’ concerning critical exponents of percolation in several two-dimensional lattices. In particular, our main purpose is to clarify how universal is ‘‘universality.’’ For this purpose, we choose the...
Ausführliche Beschreibung

By making use of finite-size scaling and Monte Carlo simulations, we study the so-called ‘‘universality’’ concerning critical exponents of percolation in several two-dimensional lattices. In particular, our main purpose is to clarify how universal is ‘‘universality.’’ For this purpose, we choose the following five lattices–square and kagome$iaa— (both periodic where the number z of nearest-neighbor sites is single valued, i.e., z=4), dice (periodic where z is mixed valued, i.e., z=3 and 6, the average z¯ being four), Penrose tiling (nonperiodic where z is mixed valued, i.e., z=3, 4, 5, 6, and 7, the average z¯ being four), and the dual lattice of Penrose (nonperiodic where z is single valued, i.e., z=4). For both site and bond percolation of these lattices, we analyze the results of our Monte Carlo simulations and evaluate six critical exponents, all of which are in good agreement with respective values predicted theoretically. Our results indicate that ‘‘universality’’ is really universal irrespective of classes of problems, i.e., whether bond or site; irrespective of kinds of lattices, i.e., whether periodic or nonperiodic; and irrespective of types of coordination, i.e., whether single valued or mixed valued. Ausführliche Beschreibung