Equation of state for the infinite cluster and backbone in the anisotropic square lattice.

Abstract

We use a real-space renormalization-group procedure recently developed for calculating equations of state for geometrical problems, to treat bond percolation in the anisotropic square lattice. By choosing a convenient self-dual cluster, we calculate, for all values of the occupancy probabilities ${p}_{x}$ and ${p}_{y}$ (along the x and y axes, respectively), the order parameters ${P}_{\ensuremath{\infty}}$(${p}_{x}$,${p}_{y}$) and ${P}_{\ensuremath{\infty}}^{B}$(${p}_{x}$,${p}_{y}$), respectively, associated with the complete percolating infinite cluster and with its backbone. An interesting difference appears between these two quantities whenever one of the occupancy probabilities, for example ${p}_{y}$, equals unity: ${\mathrm{lim}}_{{\mathrm{p}}_{\mathrm{y}}\ensuremath{\rightarrow}1}$${\mathrm{P}}_{\mathrm{\ensuremath{\infty}}}$(${\mathrm{p}}_{\mathrm{x}}$,${p}_{y}$) is discontinuous at ${p}_{x}$=0 (where ${P}_{\ensuremath{\infty}}$ jumps from 0 to 1), whereas ${\mathrm{lim}}_{{\mathrm{p}}_{\mathrm{y}}\ensuremath{\rightarrow}1}$${\mathrm{P}}_{\mathrm{\ensuremath{\infty}}}^{\mathrm{B}}$(${\mathrm{p}}_{\mathrm{x}}$,${p}_{y}$) continuously increases from 0 to 1 when ${p}_{x}$ increases from 0 to 1. Through a convenient extrapolation procedure which includes the use of the best available values for the critical exponents \ensuremath{\beta} and ${\ensuremath{\beta}}^{B}$, we obtain values for ${P}_{\ensuremath{\infty}}$ and ${P}_{\ensuremath{\infty}}^{B}$ which are believed to be numerically quite reliable. In particular, ${P}_{\ensuremath{\infty}}$(p,p)\ensuremath{\sim}A(p${\mathrm{\ensuremath{-}}(1/2))}^{\ensuremath{\beta}}$ (\ensuremath{\beta}=(5/36) and A\ensuremath{\simeq}1.25) and ${P}_{\ensuremath{\infty}}^{B}$(p,p)\ensuremath{\sim}${A}^{B}$(p-(1/2)) ${\mathrm{\ensuremath{\beta}}}^{\mathrm{B}}$ (${\ensuremath{\beta}}^{B}$\ensuremath{\simeq}0.53 and ${A}^{B}$\ensuremath{\simeq}1.92).