Federico Camia (Abu Dhabi), Dmitry Chelkak (Paris), Clément Hongler (Lausanne), Jesper Jacobsen (Paris), Vieri Mastropietro (Milano), Slava Rychkov (Bures).

Alessandro Giuliani (Roma): Conformal invariance and Renormalization Group.

Federico Camia (Abu Dhabi): Planar Ising model and Euclidean field theory. pdf

The Ising model, introduced by Lenz in 1920 to describe ferromagnetism, is one the most studied models of statistical mechanics. Its two dimensional version has played a special role in rigorous statistical mechanics since Peierls' proof of a phase transition in 1936 and Onsager's derivation of the free energy in 1944. This continues to be the case today, thanks to new results that make the connection between the planar Ising model and the corresponding Euclidean field theory increasingly more explicit. In this talk I will introduce the Ising model and discuss recent results on its critical and near-critical scaling limits. I will focus in particular on the scaling behavior of the magnetization field and on the presence of a mass gap in the corresponding near-critical Euclidean field theory. (Based on joint work with R. Conijn, C. Garban, J. Jiang, D. Kiss, and C.M. Newman.)

Dmitry Chelkak (Paris): Bipartite dimer model: perfect t-embeddings and convergence to the GFF.

We discuss a concept of `perfect t-embeddings’, or `p-embeddings', of weighted bipartite planar graphs. (T-embeddings also appeared under the name Coulomb gauges in a recent work of Kenyon, Lam, Ramassamy and Russkikh; in particular they generalize s-embeddings of planar graphs carrying the Ising model.) From our perspective, p-embeddings provide a relevant framework for the convergence (to an appropriate GFF) of the dimer height function fluctuations on big planar graphs; the long-term perspective is to attack the Ising and the bipartite dimer models on random planar maps in this way. To support this approach, we provide several results in deterministic setups, recently obtained in collaborations with Benoît Laslier, Sanjay Ramassamy and Marianna Russkikh.

Clément Hongler (Lausanne): Ising Model and Conformal Field Theory.

In this talk, we will review some recent results on the rigorous connection between the Ising model and Conformal Field Theory and discuss how this connection can be extended for other models.

Jesper Jacobsen (Paris): Four-point functions in the Fortuin-Kasteleyn cluster model. pdf

The determination of four-point correlation functions of two-dimensional lattice models is of fundamental importance in statistical physics. In the limit of an infinite lattice, this question can be formulated in terms of conformal field theory (CFT). For the so-called minimal models the problem was solved more than 30 years ago, by using that the existence of singular states implies that the correlation functions must satisfy certain differential equations. This settles the issue for models defined in terms of local degrees of freedom, such as the Ising and 3-state Potts models. However, for geometrical observables in the Fortuin-Kasteleyn cluster formulation of the Q-state Potts model, for generic values of Q, there is in general no locality and no singular states, and so the question remains open. As a warm-up to solving this problem, we discuss which states propagate in the s-channel of such correlation functions, when the four points are brought together two by two. To this end we combine CFT methods with algebraic and numerical approaches to the lattice model. Further insight can be gained by comparing, for particular values of Q (the so-called "root of unity" cases), the four-point functions of local operators in an associated height model (of the RSOS, or graph homomorphism type) with those defined geometrically in the Fortuin-Kasteleyn model. Crucially, those two formulations lead to subtly different weightings of the contributing configurations.

Vieri Mastropietro (Milano): Universality in Ising, Vertex, Dimers and beyond.

The critical exponents in a large class of statistical mechanics models are non trivial functions of the parameters and depend on all the microscopic details. It has been conjectured that such exponents verify exact universality relations, which were checked in certain solvable cases. In recent times, several of such universality relations have been rigorously proved in a wide class of systems, including non solvable ones. The proof is based on a combination of rigorous renormalization Group methods, allowing to write the exponents in the form of convergent series, combined with exact and emerging Ward Identities. The role of the irrelevent terms is essential in the proof. Similar methods can be applied to a varity of other systems, ranging from QFT to condensed matter.

Slava Rychkov (Bures): 3D Ising model: a view from the (conformal bootstrap) island.

I will describe the bootstrap CFT axioms, and give an informal introduction to what has been learned about the critical 3d Ising model using these axioms.