We provide a uniformly-positive point-wise lower bound for the two-point function of the classical spin O(N ) model on the torus of Z d , d ≥ 3, when N ∈ N >0 and the inverse temperature β is large enough. This is a new result when N > 2 and extends the classical result of Fröhlich, Simon and Spencer (1976). Our bound follows from a new site-monotonicity property of the two-point function which is of independent interest and holds not only for the spin O(N ) model with arbitrary N ∈ N >0 , but for a wide class of systems of interacting random walks and loops, including the loop O(N ) model, random lattice permutations, the dimer model, the double-dimer model, and the loop representation of the classical spin O(N ) model. *

We consider an extension of the interchange process on the complete graph, in which a fraction of the transpositions are replaced by 'reversals'. The model is motivated by statistical physics, where it plays a role in stochastic representations of xxz-models. We prove convergence to PD( 1 2 ) of the rescaled cycle sizes, above the critical point for the appearance of macroscopic cycles. This extends a result of Schramm on convergence to PD(1) for the usual interchange process.

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 $$N > 1$$ N > 1 . Our result is new when $$N > 3$$ N > 3 , in which case no Lee–Yang theorem is available, it is an alternative to Lee–Yang when $$N = 2, 3$$ 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.

We look at the general SU(2) invariant spin-1 Heisenberg model. This family includes the well known Heisenberg ferromagnet and antiferromagnet as well as the interesting nematic (biquadratic) and the largely mysterious staggered-nematic interaction. Long range order is proved using the method of reflection positivity and infrared bounds on a purely nematic interaction. This is achieved through the use of a type of matrix representation of the interaction making clear several identities that would not otherwise be noticed. Using the reflection positivity of the antiferromagnetic interaction one can then show that the result is maintained if we also include an antiferromagnetic interaction that is sufficiently small. * b.lees@warwick.ac.uk