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Balaban has developed a very powerful renormalization group method for analyzing lattice quantum field theories. This paper by J. Dimock is the first of three giving an expository account of Balaban's approach to the renormalization group, illustrating the method with a treatment of the the ultraviolet problem for the scalar $\Phi^4$ model on a toroidal lattice in dimension $d=3$.
This third part demonstrates convergence of the expansion and completes the proof of a stability bound. See also part I and part II.