Random matrix Hamiltonians and their curious mapping to a particle hopping on a hyperbolic lattice

I will highlight some concepts from free probability theory and a few applications in quantum many-body dynamics. Mainly, I will focus on our recent work which deals with the physics of random matrix Hamiltonians that obey spatial locality of interaction in 1D. Equilibrium physics and energy transport in such models will be discussed from the point of view of a curious mapping to single particle hopping on hyperbolic lattices, governed by certain Coxeter reflection groups. I will outline what outstanding questions remain and where I hope to go with such models in the future.