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

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 $n \in [-2,2).$