  Starting from the N-particle Nelson Hamiltonian defined by imposing an
ultraviolet cutoff, we perform ultraviolet renormalization by showing that in
the zero cutoff limit a self-adjoint operator exists after a logarithmically
divergent term is subtracted from the original Hamiltonian. We obtain this term
as the diagonal part of a pair interaction appearing in the density of a Gibbs
measure derived from the Feynman-Kac representation of the Hamiltonian. Also,
we show existence of a weak coupling limit of the renormalized Hamiltonian and
derive an effective Yukawa interaction potential between the particles.
