**Abstract:**
In this talk (2*50'), I will start by introducing spin systems on Z^d with continuous symmetry and I will discuss a new way to prove continuous symmetry breaking for any dimension d\geq 3 which does not rely on the standard tool for this known as "reflection positivity".
Our method applies to models whose spins take values in S^1, SU(n) or SO(n) in the presence of a certain quenched disorder called the Nishimori line. The proof of continuous symmetry breaking is based on two ingredients
1) the notion of "group synchronization" in Bayesian statistics. In particular a recent result by Abbe, Massoulié, Montanari, Sly and Srivastava (2018) which proves group synchronization when d\geq 3.
2) a gauge transformation on both the disorder and the spin configurations which goes back to Nishimori (1981).
I will end the talk with an application of these techniques to a deconfining transition for U(1) lattice gauge theory on the Nishimori line.
This is a joint work with Tom Spencer (https://arxiv.org/abs/2109.01617).