We have studied the magnetic-field-driven quantum phase transitions in Josephson junction arrays with a large coordination number. The characteristic energies were extracted in both the superconducting and insulating phases by integrating the current
-voltage characteristics over a voltage range 2eVleqk_B T. For the arrays with a relatively strong Josephson coupling, we observed duality between the energies in the superconducting and insulating phases. The arrays with a weaker Josephson coupling demonstrate an intermediate, bad metal regime in weak magnetic fields; this observation underlines the importance of vortex pinning at large scales and, presumably, emergent inhomogeneity in the presence of strong offset charge disorder.
