[STAT/AP] Aernout van Enter: Dyson models with random boundary conditions

06 November 2023 15:45 till 16:45 - Location: Lecture Hall G | Add to my calendar

I discuss the low-temperature behaviour of Dyson models (polynomially decaying long-range Ising models in one dimension) in the presence of random boundary conditions. For typical random (i.i.d.) boundary conditions Chaotic Size Dependence occurs, that is, the pointwise thermodynamic limit of the finite-volume Gibbs states for increasing volumes does not exist, but the sequence of states moves between various possible limit points. As a consequence it makes sense to study distributional limits, the so-called "metastates" which are measures on the possible limiting Gibbs measures.

The Dyson model is known to have a phase transition for decay parameters α between 1 and 2. We show that the metastate obtained from random boundary conditions changes character at α =3/2. It is dispersed in both cases, but it changes between being supported on two pure Gibbs measures when α is less than 3/2 to being supported on mixtures thereof when α is larger than 3/2.

Joint work with Eric Endo (NYU Shanghai) and Arnaud Le Ny (Paris-Est)

We also discuss the relation with a recent high-temperature result by Johansson Oberg and Pollicott about regularity of eigenfunctions of Transfer Operators. ( work in progress with Evgeny Verbitskiy and Mirmukshin Makhmudov).