TITLE: Spin correlations in the planar Ising model and orthogonal
polynomials: a direct link via fermionic observables.
ABSTRACT:
The famous explicit formulae for the full-plane (diagonal and row)
spin-spin correlations in the Ising model via Toeplitz determinants
come back to the work of Kaufmann and Onsager in 1940's. Notably,
those results had a big influence on the theory of orthogonal
polynomials itself, motivating Szego to prove a strong form of his
classical theorem on asymptotics of Toeplitz determinants.
Nevertheless, to the best of our knowledge, no direct construction of
the corresponding orthogonal polynomials in terms of the Ising model
was known until recently, and Toeplitz determinants appeared per se as
a result of some algebraic or combinatorial computations.
Using the language of fermionic observables developed by Smirnov and
others recently, we show that one can build a sequence of polynomials
using the values of spinor fermionic observable at lattice points, and
rephrase the massive-holomorphicity property of the observable as the
orthogonality condition for those polynomials. Thus, we give a new
simple derivation of the explicit formulae for spin-spin correlations
in the full plane. At the same time, our methods can be applied in
other setups (e.g., to compute a magnetization in the half-plane),
leading to a number of explicit formulae unknown before.
Based on a joint work with Clement Hongler (in progress).