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