We consider a class of Lévy-type processes on which spectral analysis technics can be made to produce optimal results, in particular for the decay rate of their survival probability and for the spectral gap of their ground state transform. This class is defined by killed symmetric Lévy processes under general random time-changes satisfying some integrability assumptions. Our results reveal a connection between those processes and a family of spin systems. This connection, where the free energy and correlations play an essential role, is, up to our knowledge, new, and relates some key properties of objects from the two families. When the underlying Lévy process is a Brownian motion, the associated spin system turns out to have interactions of a rather nice form that are a natural alternative to the quadratic interactions from the lattice Gaussian Free Field. More general Lévy processes give rise to more general interactions in the associated spin systems.