Sarnak, Peterde Courcy-Ireland, MatthewMathematics Department2018-10-092026-09-302018-10-092026-09-302018http://arks.princeton.edu/ark:/88435/dsp01x920g0628https://theses-dissertations.princeton.edu/handle/88435/dsp01x920g0628We study the monochromatic ensemble of random functions in the generality of a compact Riemannian manifold of any dimension. We prove equidistribution of local integrals at scales within a logarithmic factor of the optimal wave scale. On the two-dimensional sphere, we prove a limit theorem for the distribution of these integrals. We also study nodal domains, giving explicit (but embarrassing) lower bounds for the Nazarov-Sodin constant in dimension 2 and 3 and an estimate of the high-dimensional behaviour.enMathematicsAcademic dissertations (Ph.D.)