Exponential decay of correlations for O(N) spin systems for arbitrary N

VIENNA PROBABILITY SEMINAR

The Spin O(N) model is a classical statistical mechanics model whose configurations are collections of unit vectors taking values on the surface of a N -1 dimensional unit sphere. Some special cases are the Ising model (N = 1), the XY model (N = 2), and the classical Heisenberg model (N = 3). Despite the fact that it is a very classical model, there remain important gaps in understanding, particularly in the case N > 2. This talk will present a new recent result about exponential decay of correlations for arbitrary (non-zero) values of the external magnetic field and arbitrary spin dimension N>1, extending previous results which hold only for N=1,2,3. Our proof is probabilistic and employs a representation of the model as a system of coloured random paths which is of independent interest. The talk is based on a joint work with B. Lees.