  We consider the renormalized relativistic Nelson model in two spatial
dimensions for a finite number of spinless, relativistic quantum mechanical
matter particles in interaction with a massive scalar quantized radiation
field. We find a Feynman-Kac formula for the corresponding semigroup and
discuss some implications such as ergodicity and weighted $L^p$ to $L^q$
bounds, for external potentials that are Kato decomposable in the suitable
relativistic sense. Furthermore, our analysis entails upper and lower bounds on
the minimal energy for all values of the involved physical parameters when the
Pauli principle for the matter particles is ignored. In the translation
invariant case (no external potential) these bounds permit to compute the
leading asymptotics of the minimal energy in the three regimes where the number
of matter particles goes to infinity, the coupling constant for the
matter-radiation interaction goes to infinity and the boson mass goes to zero.
