Quantum $k$-core Percolation on the Bethe Lattice

Quantum percolation is the study of hopping transport of a quantum particle on randomly diluted percolation clusters. We investigate the Landauer conductance through the dilute Bethe lattice. We show that (1) $p_q$, the quantum percolation critical probability, is greater than $p_c$, the geometric percolation critical probability, and (2) for $p_q<1$ that the quantum conductance transition is continuous with a quantum conductance exponent of 2, as in the classical case. We also study the Landauer conductance through a dilute Bethe lattice where the dilution is subject to the condition that each occupied bond/site must have at least $k$ occupied neighboring bonds/sites. This geometric constraint defines $k$-core percolation. We find, again, that $p_q>p_c$ and, for $p_q<1$, we calculate a quantum conductance exponent of 2 for $k=3$ and a coordination number of four.