Massless Latticeφ44Theory: A Nonperturbative Control of a Renormalizable Model

We describe a rigorous argument establishing infrared asymptotic freedom and other related properties (e.g., scaling with logarithmic corrections) of the weakly coupled massless lattice ${\phi }^4$ model in four Euclidean dimensions. Our method combines the block-spin renormalization-group analysis with the technique of convergent high-temperature expansions. It should also extend to the ultraviolet asymptotically free models such as the negative-coupling continuum ${\phi }_4^4$ theory.