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Exponential decay of transverse correlations for spin systems with continuous symmetry and non-zero external field

2020, Lees, Benjamin, Taggi, Lorenzo

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.

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Macroscopic loops in the Bose gas, Spin O(N) and related models

2022, Quitmann, Alexandra, Taggi, Lorenzo

We consider a general system of interacting random loops which includes several models of interest, such as the textitSpin O(N) model, textitrandom lattice permutations, a version of the textitinteracting Bose gas in discrete space and of the textitloop O(N) model. We consider the system in ℤd, d ≥ 3, and prove the occurrence of macroscopic loops whose length is proportional to the volume of the system. More precisely, we approximate ℤd by finite boxes and, given any two vertices whose distance is proportional to the diameter of the box, we prove that the probability of observing a loop visiting both is uniformly positive. Our results hold under general assumptions on the interaction potential, which may have bounded or unbounded support or introduce hard-core constraints.