We discuss uniform small-scale sign balance properties of random Laplace eigenfunctions, including random spherical harmonics and band-limited random waves on smooth Riemannian manifolds. Our main result is that random eigenfunctions are sign-balanced above a precisely determined scale with almost full probability. The scale is proven to be optimal up to a logarithmic power of the energy. Extending the notion of balance to arbitrary levels, we determine the precise optimum scale above which random eigenfunctions are uniformly volume-balanced with respect to non-zero levels. Beyond their intrinsic interest, our results serve as a model for a natural conjecture on the optimal scale at which deterministic Laplace eigenfunctions are sign balanced. The talk is based on joint work with Igor Wigman.
