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

TL;DR

This paper presents an exact solution for bond-percolation thresholds with inhomogeneous probabilities on the square lattice using duality with real-space renormalization, offering a more straightforward approach than previous methods.

Contribution

It introduces a novel, simplified formulation for calculating bond-percolation thresholds using duality and real-space renormalization, extending applicability beyond the square lattice.

Findings

Exact bond-percolation thresholds derived for the square lattice.

Formulas applicable to other lattice types for threshold estimation.

Method improves upon previous approaches in simplicity and generality.

Abstract

We obtain the exact solution of the bond-percolation thresholds with inhomogenous 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 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 from our derivation can give several estimations for the bond-percolation thresholds on other lattices rather than the square lattice.