Self-avoiding walk, spin systems, and renormalization

TL;DR

This paper reviews mathematical challenges in self-avoiding walks and spin systems, highlighting recent progress achieved through a rigorous nonperturbative renormalization group approach.

Contribution

It introduces a rigorous nonperturbative renormalization group method to advance understanding of critical phenomena in these models.

Findings

Progress in understanding critical exponents

Development of nonperturbative RG techniques

Insights into mathematical properties of spin systems

Abstract

The self-avoiding walk, and lattice spin systems such as the $\varphi^4$ 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.