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

A real space renormalization group procedure recently developed for calculating equations of states for geometrical problems is used, to treat bond percolation in the anisotropic square lattice. By choosing a convenient self-dual cluster, for all values of the occupancy probabilities P sub(x) and P sub(y) (along the x and y axes respectively), the order parameters P infinity (P sub(x),P sub(y)) and P sup(B) infinity (P sub(x),P sub(y)) respectively associated with the complete percolating infinite cluster and with its backbone are calculated. An interesting difference appears between these two quantities whenever one of the occupancy probabilities, say P sub(y), equals unity: lim sub(P sub(y) → l) P infinity (P sub(x),P sub(y) is discontinuous at P sub(x)=0 (where P sub(infinity) jumps from 0 to 1), whereas lim sub(P sub(y) → 1) P sup(B) sub(infinity) (P sub(x),P sub(y)) continuously increases from 0 to 1 when P sub(x) increases from 0 to 1. Through a convenient extrapolation procedure which includes the use of the best available values for the critical exponents β and β sup(B), values for P sub(infinity) and P sup(B) sub(infinity) which are believed to be numerically quite reliable are obtained. In particular, P sub(infinity) (p,p) approx. A (p-1/2) sup(β) (β=5/36 and A approx. 1.25) and P sup(B) sub(infinity) (p,p) approx. A sup(B) (p-1/2) sup(β) sup(B) (β sup(B) approx. 0.53 and A sup(B) approx. 1.92). (Author)