Séminaire de Mathématique

A hierarchical model for three-dimensional conformal probability

by Prof. Abdelmalek Abdesselam (University of Virginia & IHES)

Salle du Conseil (IHES)

Salle du Conseil


Le Bois Marie 35, route de Chartres 91440 Bures-sur-Yvette

I will introduce a hierarchical model for a Euclidean conformal field theory in three dimensions. This is a real valued distributional random field over Q_p^3 (instead of R^3). However, I will not assume any knowledge of p-adics. The model is a scalar phi-four theory obtained as a scaling limit of a fixed critical ferromagnetic Gibbs random field on the unit lattice. This is analogous to the scaling limit of the 2d Ising model studied recently by Dubedat, Camia, Garban, Newman, Chelkak, Hongler and Izyurov. I will review joint work with Ajay Chandra and Gianluca Guadagni which constructed not only the random field itself (the spin field) but also its pointwise square (energy field). This is based on a new rigorous renormalization group method whose main feature is the ability to handle space-dependent couplings.

The square field exhibits an anomalous scaling dimension as predicted by Wilson more than 40 years ago. This is the first rigorous construction by renormalization group methods of a bosonic field with anomalous scaling. The key to this property is a new result in dynamical system theory which is an infinite-dimensional generalization of the Poincare-Koenigs holomorphic linearization theorem.