# Nodal lines for random eigenfunctions of the Laplacian on the torus

05/24/2007 - 11:00
05/24/2007 - 12:00
Speaker:
Igor Wigman (CRM and McGill)
Location:
CRM, Room 5340
Abstract:
We study the volume of nodal sets for eigenfunctions of the Laplacian on the standard torus in two or more dimensions. We consider a sequence of eigenvalues $\normalsize 4\pi^2\lambda$ with growing multiplicity $\normalsize N\to\infty$, and compute the expectation and variance of the volume of the nodal set with respect to a Gaussian probability measure on the eigenspaces. We show that the expected volume of the nodal set is $\normalsize const\sqrt{\lambda}$. Our main result is that the variance of the volume normalized by $\normalsize \sqrt{\lambda}$ is bounded by $\normalsize O(1/\sqrt{N})$, so that the normalized volume has vanishing fluctuations as we increase the dimension of the eigenspace.
Last edited by on Tue, 05/22/2007 - 10:10