One of the main goals of statistical physics is to observe how spins displayed along a lattice Z^d interact together and fluctuate. When the spins belong to a discrete set (for example the celebrated Ising model where spins \sigma_x belong to {-1,+1}), the nature of the phase transitions which arise as one varies the temperature is now rather well understood. When the spins belong instead to a continuous space (for example the unit circle S^1 for the so-called XY model, the unit sphere S^2 for the classical Heisenberg model etc.), the nature of the phase transitions differs drastically from the discrete symmetry setting. The case where the (continuous) symmetry is non-Abelian is currently more mysterious than when the symmetry is Abelian. In the later case, phase transitions are caused by a change of behaviour of certain monodromies in the system called « vortices ». They are called topological phase transitions for this reason.
In this talk, after an introduction to the mathematics of spin systems with a continuous symmetry, I will present some recent results on these spins systems. One proof will happen to rely on an intriguing Bayesian statistics problem (!).
The talk will not require any background in statistical physics/probability and will be based on joint works with Juhan Aru, Paul Dario, Avelio Sepúlveda and Tom Spencer.