Conducting backbone in percolating Bethe lattices

An exact theory is given for the backbone fraction, i.e. the percolation fraction minus the fraction of tag ends, of a Bethe lattice with a fraction p of conducting bonds randomly added in a bond or site process. In either the bond or site problem the critical exponent of the backbone fraction near the percolation threshold is two, a value twice the critical exponent of the percolation fraction and equal to the critical exponent of the conductivity fraction of a Bethe lattice. Previously confusion between the percolation probability and the percolation fraction (density of infinite clusters) is discussed.