  We consider mass concentration properties of Laplace eigenfunctions
$\varphi_\lambda$, that is, smooth functions satisfying the equation $-\Delta
\varphi_\lambda = \lambda \varphi_\lambda$, on a smooth closed Riemannian
manifold. Using a heat diffusion technique, we first discuss mass
concentration/localization properties of eigenfunctions around their nodal
sets. Second, we discuss the problem of avoided crossings and (non)existence of
nodal domains which continue to be thin over relatively long distances.
Further, using the above techniques, we discuss the decay of Laplace
eigenfunctions on Euclidean domains which have a central "thick" part and
"thin" elongated branches representing tunnels of sub-wavelength opening.
Finally, in an Appendix, we record some new observations regarding sub-level
sets of the eigenfunctions and interactions of different level sets.
