  We use probabilistic tools based on Brownian motion and Feynman-Kac formulae
to investigate the heat profile for the ground state Dirichlet and second
Neumann eigenfunctions. Among other topics, we comment on supremum norm bounds
for ground state Dirichlet eigenfunctions and look at the corresponding Neumann
problem, namely the comparison of maximum temperatures on the interior and the
boundary, the latter being partially motivated by the hot spots problem. We
also investigate the proximity/distance of level sets of ground state Dirichlet
eigenfunctions, some with analogous statements for Neumann eigenfunctions.
Domains with bottlenecks make occasional appearances as an illuminating example
as well as testing ground for our theory.
