Large time behaviour of heat semigroups (and more generally, of positive
selfadjoint semigroups) is studied. Convergence of the semigroup to the ground
state and of averaged logarithms of kernels to the ground state energy is shown
in the general framework of positivity improving selfadjoint semigroups. This
framework includes Laplacians on manifolds, metric graphs and discrete graphs.Comment: 24 page
