Scaling Violation in O(N) Vector Models

TL;DR

This paper studies $O(N)$-symmetric vector models in the double scaling limit, revealing divergences in free energy that relate to branched polymers and string-like behaviors in various dimensions.

Contribution

It provides explicit forms of residual divergences in the free energy of $O(N)$ models, connecting vector models to branched polymers and string theories.

Findings

Residual divergences in free energy are characterized.

Models interpolate between Cayley trees and random walks.

Connections to string theory in one dimension are established.

Abstract

We investigate $O(N)$-symmetric vector field theories in the double scaling limit. Our model describes branched polymeric systems in $D$ dimensions, whose multicritical series interpolates between the Cayley tree and the ordinary random walk. We give explicit forms of residual divergences in the free energy, analogous to those observed in the strings in one dimension.