Exponential decay of transverse correlations for O(N) spin systems and related models

We prove exponential decay of transverse correlations in the Spin O(N) model for arbitrary non-zero values of the external magnetic field and arbitrary spin dimension N>1. Our result is new when N>3, , in which case no Lee–Yang theorem is available, it is an alternative to Lee–Yang when N=2,3, and also holds for a wide class of multi-component spin systems with continuous symmetry. The key ingredients are a representation of the model as a system of coloured random paths, a ‘colour-switch’ lemma, and a sampling procedure which allows us to bound from above the ‘typical’ length of the open paths.