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

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