Level spacing statistics for quantum $k$-core percolation

Quantum percolation is the study of hopping transport of a quantum particle on randomly diluted percolation clusters. Quantum $k$-core percolation is the study of quantum transport on $k$- core percolation clusters where each occupied bond must have at least $k$ occupied neighboring bonds. Within the random phase approximation, we found a random first-order phase transition for the $k$-core conduction transition on the Bethe lattice, and $p_q$, the quantum percolation critical probability, is equal to $p_c$, the geometric percolation critical probability [Phys. Rev. B {\bf 82},104211 (2010)]. To further test this result, we numerically compute the level spacing distribution as a function of occupation probability $p$ and system size. The simulation results provide confirmation for the existence of a discontinuous onset of quantum conduction at $p_q=p_c$.