Statistical Mechanics of Self-Avoiding Manifolds (Part II)

TL;DR

This paper develops a rigorous theoretical framework for analyzing self-avoiding manifolds using statistical mechanics, establishing renormalizability and extending field theory methods to non-local interactions.

Contribution

It introduces a new intrinsic distance geometry approach and proves renormalizability of the model to all orders, extending field theory techniques to non-local interactions.

Findings

Established one-loop renormalizability of the model.

Extended renormalization to all orders.

Developed a multi-local operator product expansion.

Abstract

We consider a model of a D-dimensional tethered manifold interacting by excluded volume in R^d with a single point. Use of intrinsic distance geometry provides a rigorous definition of the analytic continuation of the perturbative expansion for arbitrary D, 0 < D < 2. Its one-loop renormalizability is first established by direct resummation. A renormalization operation R is then described, which ensures renormalizability to all orders. The similar question of the renormalizability of the self-avoiding manifold (SAM) Edwards model is then considered, first at one-loop, then to all orders. We describe a short-distance multi-local operator product expansion, which extends methods of local field theories to a large class of models with non-local singular interactions. It vindicates the direct renormalization method used earlier in part I of these lectures, as well as the corresponding scaling laws.