Phase Diagram of the Quantum Random Energy Model

Abstract

We prove Goldschmidt’s formula (Goldschmidt in Phys Rev B 47:4858–4861, 1990) for the free energy of the quantum random energy model. In particular, we verify the location of the first order and the freezing transition in the phase diagram. The proof is based on a combination of variational methods on the one hand, and bounds on the size of percolation clusters of large-deviation configurations in combination with simple spectral bounds on the hypercube’s adjacency matrix on the other hand.