Duality with real-space renormalization and its application to bond percolation

We obtain the exact solution of the bond-percolation thresholds with inhomogeneous probabilities on the square lattice. Our method is based on the duality analysis with real-space renormalization, which is a profound technique invented in the spin-glass theory. Our formulation is a more straightforward way than that of a very recent study on the same problem [R. M. Ziff et al., J. Phys. A: Math. Theor. 45, 494005 (2012)]. The resultant generic formulas from our derivation can give several estimations for the bond-percolation thresholds on other lattices rather than the square lattice.

Figure 1

The unit cell of the inhomogeneous bond-percolation problem on the square lattice. The unit cell is a part of the square lattice, which is bounded by the dashed lines. The assigned values $p$, $r$, $s$, and $t$ are the probabilities to connect both ends on each bond on the unit cell.

Figure 2

Duality relation of the bond-percolation problem. The dotted lines denote the disconnected bonds. The bold lines represent the connected bonds. The white circles denote the original sites. The black circles represent the dual sites (original plaquettes).

Figure 3

The inhomogeneous bond-percolation problem on the triangular and hexagonal lattices. The assigned values $p$, $r$, and $s$ are the connected probabilities assigned on each bond on the unit cell. On the hexagonal lattice, we put the dual probabilities $p^{*}$, $r^{*}$, and $s^{*}$, which are obtained after the dual transformation. The black circle on the hexagonal lattice represents the internal site we sum over in the star-triangle transformation.

Figure 4

The unit cell with four bonds for the inhomogeneous case on the square lattice. The black circles denote the internal spins that we sum over, while the white ones are fixed as ${\varphi }_i=0$.

Figure 5

The cluster for the duality with real-space renormalization on the square lattice. The white circles denote the fixed spins for evaluation of the principal Boltzmann factors. The black circles express the spins we trace over (similar to the star-triangle transformation) to obtain the principal Boltzmann factors.

Figure 6

Transformations into the hexagonal and triangular lattices from the square lattice.

Figure 7

Two four-terminal unit cells. The shaded squares express the unit cells. The left panel denotes the original square lattice, and the right one depicts the dual lattice.

Figure 8

The bow-tie lattice. The shaded squares express the four-terminal unit cells.

Figure 9

The Kagomé lattice. The four-terminal unit cell consists of the up-pointing and down-pointing triangles.