School of Mathematical Sciences

Ising model on planar graphs: s-holomorphic functions and embeddings. menu

Ising model on planar graphs: s-holomorphic functions and embeddings.

Speaker: 
Dmitry Chelkak (ENS, Paris)
Date/Time: 
Wed, 18/10/2017 - 13:00
Room: 
Queen's Building, W316

During the last decade, a significant progress in the understanding of the critical Ising model on nice 2D lattices has been achieved, basing on the careful analysis of the so-called s-holomorphic observables (aka lattice fermions). Surprisingly and embarrassingly, despite the facts that the rigid structure of s-holomorphic functions exists on every weighted planar graph and that the conformally invariant behavior arising in the scaling limit should be very universal, the existing proofs of convergence results highly rely on some particular trick (sub/super-harmonicity of the primitives of f^2), which works only for the special case of isoradial graphs, with prescribed Ising weights. The main purpose of this talk is to discuss what can be done in more general settings: from some explicit computations for the "layered" model in the half-plane (unpublished work with Clement Hongler (Lausanne)) to a new embedding of generic weighted planar graphs into the plane which might pave a way to true universality results for the critical Ising model.