Abstract: The spin 
model is an important statistical mechanics model whose spins are vectors taking value in a 
dimensional sphere. The rigorous understanding of such a model is still quite unsatisfactory, especially in the 
case. This talk will introduce a random loop representation the model, which is an intriguing mathematical object on its own and includes other important models, such as the dimer model, the double dimer model, the loop model. This representation will be used to prove new results about exponential decay of correlations in the presence of an external magnetic field which are valid for any value of 
, extending previous results
In the first talk we will introduce the spin model as well as a general loop model. For a particular choice of parameters this loop model can be connected to the spin model (as well as the loop model, the dimer and double dimer model, spatial random permutations…). By considering lengths of walks in this loop model we can prove exponential decay of correlations in the spin model.
In the second talk we will see how the loop model and spin model with external field are connected, and then see the main steps of the proof. In particular we will present a so-called “colour switching” lemma, that allows us to write spin-spin correlations as the expectation of a simple random variable in the loop model. The proof relies only on properties of the loop model and so can be applied quite generally to (for example) inhomogeneous couplings and/or external fields, as well as other related models such as the loop model.