Connective-constant Peierls bound ($p_c\le 1-{\mu }^{-1}$)

Independent bond percolation on the square lattice satisfies $p_c\le 1-{\mu }^{-1}$, where $\mu$ is its connective constant. If $(1-p)\mu <1$, closed dual graph cycles have a summable large-length tail: their counts are bounded by a polynomial factor times the count of self-avoiding walks. Conditioning a sufficiently large finite box to be open excludes short enclosing graph cycles without changing the law of the remaining edges. With positive conditional probability no enclosing closed dual graph cycle remains, so the origin belongs to an infinite percolation cluster.