We give a delocalization estimate for eigenfunctions of the discrete Laplacian on large d + 1-regular graphs, showing that any subset of the graph supporting ǫ of the L 2 mass of an eigenfunction must be large. For graphs satisfying a mild girth-like condition, this bound will be exponential in the size of the graph.

We study joint quasimodes of the Laplacian and one Hecke operator on compact congruence surfaces, and give conditions on the orders of the quasimodes that guarantee positive entropy on almost every ergodic component of the corresponding semiclassical measures. Together with the measure classification result of [Lin06], this implies Quantum Unique Ergodicity for such functions. Our result is optimal with respect to the dimension of the space from which the quasi-mode is constructed.We also study equidistribution for sequences of joint quasimodes of the two partial Laplacians on compact irreducible quotients of H × H.

Abstract. We prove quantum ergodicity for certain orthonormal bases of L 2 (S 2 ), consisting of joint eigenfunctions of the Laplacian on S 2 and the discrete averaging operator over a finite set of rotations, generating a free group. If in addition the rotations are algebraic we give a quantified version of this result. The methods used also give a new, simplified proof of quantum ergodicity for large regular graphs.

Given any compact hyperbolic surface M, and a closed geodesic on M, we construct of a sequence of quasimodes on M whose microlocal lifts concentrate positive mass on the geodesic. Thus, the Quantum Unique Ergodicity (QUE) property does not hold for these quasimodes. This is analogous to a construction of Faure-Nonnenmacher-De Bièvre in the context of quantized cat maps, and lends credence to the suggestion that large multiplicities play a role in the known failure of QUE for certain "toy models" of quantum chaos. We moreover conjecture a precise threshold for the order of quasimodes needed for QUE to hold-the result of the present paper shows that this conjecture, if true, is sharp.

We prove upper bounds on the L p norms of eigenfunctions of the discrete Laplacian on regular graphs. We then apply these ideas to study the L p norms of joint eigenfunctions of the Laplacian and an averaging operator over a finite collection of algebraic rotations of the 2-sphere. Under mild conditions, such joint eigenfunctions are shown to satisfy for large p the same bounds as those known for Laplace eigenfunctions on a surface of non-positive curvature.