with Dmitry Chelkak and Clément Hongler

We discuss the magnetization $M_m$ in the $m$-th column of the zig-zag layered 2D Ising model on a half-plane using Kadanoff-Ceva fermions and orthogonal polynomials techniques. Our main result gives an explicit representation of $M_m$ via $m×m$ Hankel determinants constructed from the spectral measure of a certain Jacobi matrix which encodes the interaction parameters between the columns. We also illustrate our approach by giving short proofs of the classical Kaufman-Onsager-Yang and McCoy-Wu theorems in the homogeneous setup and expressing $M_m$ as a Toeplitz+Hankel determinant for the homogeneous sub-critical model in presence of a boundary magnetic field.

with Dmitry Chelkak and Konstantin Izyurov

We prove universality of spin correlations in the scaling limit ofthe planar Ising model on isoradial graphs with uniformly bounded angles and $Z$–invariant weights. Specifically, we show that in the massive scaling limit, i. e., as the mesh size $\delta $ tends to zero at the same rate as the inverse temperature goes to the critical one, the two-point spin correlations in the full plane behave as $\delta^{-\frac{1}{4}} \mathbb{E}[\sigma_{u_{1}}\sigma_{u_{2}}] \rightarrow \mathcal{C}_{\sigma}^{2} \Xi (|u1−u2|,m)$as $\delta \rightarrow 0 $,where the universal constant $\mathcal{C}_{\sigma}$ and the function $\Xi (|u1−u2|,m)$ are independent of the lattice. The mass $m$ is defined by the relation $k′−1\sim 4m\delta $, where $k′$ s the Baxter elliptic parameter. This includes $m$ of both signs as well as the critical case when $\Xi (r,0) = r^{-\frac{1}{4}}$. These results, together with techniques developed to obtain them, are sufficient to extend to isoradial graphs the convergence of multi-point spin correlations in finite planar domains on the square grid, which was established in a joint work of the first two authors and C. Hongler at criticality, and by S. C. Park in the sub-critical massive regime. We also give a simple proof of the fact that the infinite-volume magnetization in the $Z$–invariant model is independent of the site and of the lattice. As compared to techniques already existing in the literature, we streamline the analysis of discrete (massive) holomorphic spinors near their ramification points, relying only upon discrete analogues of the kernel $z^{-\frac{1}{2}} $ for $m= 0$ and of $z^{-\frac{1}{2}} e^{\pm 2m|z|}$ for $m \neq 0 $. Enabling the generalization to isoradial graphs and providing a solid ground for further generalizations, our approach also considerably simplifies the proofs in the square lattice setup.

with Jhih-Huang Li

We introduce Kadanoff–Ceva order-disorder operators in the quantum Ising model. This approach was first used for the classical planar Ising model and recently put back to the stage. This representation turns out to be equivalent to the loop expansion of Sminorv’s fermionic observables and is particularly interesting due to its simple and compact formulation. Using this approach, we are able to extend different results known in the classical planar Ising model, such as the conformal invariance / covariance of correlations and the energy-density, to the spin-representation of the (1+1)-dimensional quantum Ising model.

We prove RSW type crossing estimates for the FK-Ising model on general s-embeddings with uniformly bounded angles/lengths, extending a result proven by Chelkak in for ’flat’ s-embeddings.