Glassy dynamics in the hard matrix model

We introduce the hard matrix model (HMM) which consists of an orthogonal matrix exploring an energy landscape proportional to its 1-norm. In its low-temperature phase it is confined to a harmonic well around a matrix whose entries all have the same magnitude. These ground states are proportional to Hadamard matrices, which are useful in statistics and coding theory but are combinatorially hard to enumerate or even produce. Any hopes that gradient descent on the HMM would find new Hadamards are dashed by the observed numerics: beginning at moderate size, the ground state is never found from a high-temperature quench and dynamic timescales grow superexponentially with inverse temperature. We describe our progress toward understanding these phenomena and speculate on the relevance of the HMM to structural glasses.