Uniqueness and CLT for the Ground State of the Disordered Monomer-Dimer Model on $\mathbb{Z}^{d}$

TL;DR

This paper proves the uniqueness and stability of ground states in the disordered monomer-dimer model on integer lattices, and establishes a central limit theorem for ground state weights, addressing a long-standing open problem.

Contribution

It demonstrates the absence of multiple infinite-volume ground states and proves their stability, providing new insights into disordered lattice models.

Findings

No infinite volume incongruent ground states exist.

Ground states are stable under weight perturbations.

A CLT for ground state weights on growing tori is established.

Abstract

We prove that the disordered monomer-dimer model does not admit infinite volume incongruent ground states in $\mathbb{Z}^d$ 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.