We describe two works on eigenfunctions of Laplacian on manifolds/graphs.
1) Under a natural model of random eigenfunction of the Laplacian on the 2-dimensional torus, the total length of the nodal set is a random variable whose expected value is easy to calculate. We calculate the variance. An interesting feature is that the variance can have different asymptotics along difference subsequences, depending on the distribution of lattice points on corresponding circles. This is joint work with Igor Wigman and Par Kurlberg.
2) The sign of second eigenvector of the Laplacian of a weighted graph partitions the graph into two parts. In the simplest case of a line graph endowed with i.i.d. edge weights, we observed that if the edge-weights are supported away from 0, then the two parts are almost equal, but if the edge-weights can be close to zero, then the smaller of the lengths can be anywhere between 0 and 1/2. We explain this observation. This is joint work with Arvind Ayyer.