ISRMT: Magnetization in the planar Ising model and orthogonal polynomials

We will discuss a 'synthetic' proof of explicit formulae for the magnetization in the 'zig-zag layered' planar Ising model on Z^2 that is based upon an observation that the values of certain fermionic observables equal the coefficients of an orthogonal polynomial. (In the homogeneous case, this version of classical proofs dating back to Onsager, Yang and McCoy-Wu, among other simplifications, bypasses Toeplitz determinants as an intermediate step in the computations.) The main new result is a formula for the magnetizations in the m-th column of a layered model in the zig-zag half-plane via Hankel determinants constructed from the spectral measure of a certain Jacobi matrix that encodes interaction parameters between the columns. This result holds in full generality and leads to open questions on asymptotics of such determinants in several setups of interest. Based upon arXiv:1904.09168, joint work with Clément Hongler and Rémy Mahfouf.

A recording of the talk can be found at https://youtu.be/GFqcaH0JGe8