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We prove that the disordered monomer-dimer model does not admit infinite volume incongruent ground states in Zᵈ which can be obtained as a limit of finite volume ground states. Furthermore, we also prove that these ground states are stable under perturbation of the weights in a precise sense. As an application, we obtain a CLT for the ground state weight for a growing sequence of tori. Our motivation stems from a similar and long standing open question for the short range Edwards-Anderson spin glass model.
Krishnan et al. (Tue,) studied this question.