Self-avoiding walk, spin systems, and renormalization

Abstract

The self-avoiding walk, and lattice spin systems such as the ${\phi }^4$ 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 nonperturbative renormalization group method.

Minor edits. To appear in Proc. R. Soc. London, Series A

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• Assessing the polymer coil-globule state from the very first spectral modes
• Energy of the Interacting Self-Avoiding Walk at the $\theta -$point