We give an estimate of the quantum variance for $d$-regular graphs quantised with boundary scattering matrices that prohibit back-scattering. For families of graphs that are expanders, with few short cycles, our estimate leads to quantum ergodicity f
or these families of graphs. Our proof is based on a uniform control of an associated random walk on the bonds of the graph. We show that recent constructions of Ramanujan graphs, and asymptotically almost surely, random $d$-regular graphs, satisfy the necessary conditions to conclude that quantum ergodicity holds.
We calculate joint moments of the characteristic polynomial of a random unitary matrix from the circular unitary ensemble and its derivative in the case that the power in the moments is an odd positive integer. The calculations are carried out for fi
nite matrix size and in the limit as the size of the matrices goes to infinity. The latter asymptotic calculation allows us to prove a long-standing conjecture from random matrix theory.
We describe some new families of quasimodes for the Laplacian perturbed by the addition of a potential formally described by a Dirac delta function. As an application we find, under some additional hypotheses on the spectrum, subsequences of eigenfun
ctions of Seba billiards that localise around a pair of unperturbed eigenfunctions.
