  Euclidean invariant Klein-Gordon, Dirac and massive Chern-Simons field
theories are constructed in terms of a random walk with a spin factor on a
three dimensional lattice. We exactly calculate the free energy and the
correlation functions which allow us to obtain the critical diffusion constant
and associated critical exponents. It is pointed out that these critical
exponents do not satisfy the hyper-scaling relation but the scaling
inequalities. We take the continuum limit of this theory on the basis of these
analyses. We check the universality of obtained results on other lattice
structure such as triclinic lattice and body centered lattice.
