Igor Wigman: Nodal lines of random Laplace eigenfunctions

We are interested in the length of nodal lines for eigenfunctions of the Laplacian corresponding to large eigenvalues. In case of the torus or the sphere, the eigenspaces are degenerate, so that we may endow the eigenspaces with Gaussian probability measure. We study the distribution of the length of nodal lines of random eigenfunction in the corresponding ensemble.

First, using a standard technique, we compute an exact expression for the expected value of the length. Our main result concerns the variance.

This work is joint with Zeev Rudnick and Manjunath Krishnapur. Time permitting, I will also show a recent related result joint with John Toth concerning the number of open nodal lines on a generic billiard.