Off-critical parafermions and the winding angle distribution of the O($n$) model

TL;DR

This paper establishes relationships between critical exponents and the winding angle distribution in the O(n) model, using off-critical deformations and assuming conformal field theory predictions, with proofs of exponent inequalities and conjectures for various n.

Contribution

It introduces a novel off-critical deformation approach to relate critical exponents and winding angle distribution in the O(n) model, confirming predictions from conformal field theory.

Findings

Proves relationships between surface and wedge critical exponents.

Establishes inequalities for edge exponents.

Provides conjectured exponent values for n in [-2,2).

Abstract

Using an off-critical deformation of the identity of Duminil-Copin and Smirnov, we prove a relationship between half-plane surface critical exponents $\gamma_1$ and $\gamma_{11}$ as well as wedge critical exponents $\gamma_2(\alpha)$ and $\gamma_{21}(\alpha)$ and the exponent characterising the winding angle distribution of the O($n$) model in the half-plane, or more generally in a wedge of wedge-angle $\alpha.$ We assume only the existence of these exponents and, for some values of $n,$ the conjectured value of the critical point. If we assume their values as predicted by conformal field theory, one gets complete agreement with the conjectured winding angle distribution, as obtained by CFT and Coulomb gas arguments. We also prove the exponent inequality $\gamma_1-\gamma_{11} \ge 1,$ and its extension $\gamma_2(\alpha)-\gamma_{21}(\alpha) \ge 1$ for the edge exponents. We provide conjectured values for all exponents for $n \in [-2,2).$