The self-avoiding walk, and lattice spin systems such as the φ 4 model, are models of interest both in mathematics and in physics. Many of their important mathematical problems remain unsolved, particularly those involving critical exponents. We survey some of these problems, and report on recent advances in their mathematical understanding via a rigorous non-perturbative renormalization group method.
Keywords: critical exponent; renormalization; self-avoiding walk; φ4 model.