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

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