A matrix hamiltonian with quenching is proposed conjecturing that it represents the infinite volume lattice gauge theory at N = ∞ for both weak and strong couplings. At weak couplings nonperturbative configurations are identified and it is argued that they cause a phase transition. Similar nonperturbative configurations are shown to persist at infinite N raising the hope that they might be relevant to long distance phenomena.Many thanks are due to the BNL group for their warm hospitality. Discussions with M. Creutz, and M. Okawa are gratefully acknowledged. I would like to thank R. Dashen and especially E. Mottola for generously sharing their insights.