CLT for the ground state energy of the disordered monomer Dimer model on \mathbb{Z}^{d}

Runaway Seminar

Abstract

The Monomer-Dimer model is a Gibbs measure defined on the space of matchings (not necessarily perfect) on a graph. This talk will concerned the disordered version of this problem with additional environmental randomness, the weights of the matchings are themselves random variables. The ground state of the model is a measure concentrated on matchings of minimal energy, in other words the Gibbs measure at zero temperature. In this talk, I will introduce the model on the discrete torus. In joint work with Gourab Ray, we establish the uniqueness in limit of the ground state as the lattice size tends to infinity. This involves certain ergodic theoretic tools such as the Burton Keane argument. We use this uniqueness to establish a central limit theorem for the ground state energy by applying Chatterjee’s powerful method of normal approximation.