arXiv subjects

Rémy Mahfouf

Publications and source records attributed to Rémy Mahfouf.

7 recordsLinked to original sources

arXiv2026

From the planar Ising model to quasiconformal mappings

We identify the scaling limit of full-plane Kadanoff-Ceva fermions on generic, non-degenerate $s$-embeddings. In this broad setting, the scaling limits are described in terms of solutions to conjugate Beltrami equations with prescribed singularities. For the underlying Ising model, this leads to the scaling limit of the energy-energy correlations connecting (near-)critical planar Ising models to Green kernels of uniformly elliptic operators and quasiconformal mappings. For grids approximating bounded domains in the complex plane, we establish, in the scaling regime, the conformal covariance of the energy density on critical doubly periodic graphs. We complement this result with the convergence of the energy-energy correlations for smooth limiting structures $(z,\vartheta)$ viewed as spacelike surfaces the Minkowski space $\mathbb{R}^{(2,1)}$. This includes explicit formulae when $(z,\vartheta)$ is maximal, generalizing conformal covariance to new setups involving an additional change of metric. All scaling factors obtained are local and expressed in terms of the geometry of the embedding, even in situations where they vary drastically from one region to another. These results confirm the predictions of Chelkak and highlight that the scaling limits of generic (near-)critical Ising models naturally live in a substantially richer conformal structure than the classical Euclidean one.

math.PR↗

arXiv2025

Universality of Ising spin correlations on critical doubly-periodic graphs

We establish conformal invariance of Ising spin correlations on critical doubly periodic graphs, showing that their scaling limit coincides with that of the critical square lattice, as originally proved by Chelkak, Hongler and Izyurov. To overcome the absence of integrability and quantitative full plane constructions in the periodic setting, we combine discrete analytic tools with random cluster methods. This result completes the universality picture for periodic lattices, whose criticality condition was identified by Cimasoni and Duminil-Copin and whose conformal structure and interface convergence were obtained by Chelkak.

math.PR↗

arXiv2025

Crossing estimates for the Ising model on general s-embeddings

We prove Russo-Seymour-Welsh type crossing estimates for the FK-Ising model on general s-embeddings whose origami map has an asymptotic Lipschitz constant strictly smaller than $1$, provided a mild non-degeneracy assumption is satisfied. This result extends the original work of Chelkak and provides a general framework to prove that connection probabilities between boundaries of boxes remain bounded away from $0$ and $1$. It is explained that one cannot prove similar estimates without a similar assumption on the origami map, and allows to propose some notion of critical model for generic planar graphs, that can be rephrased from the perspective of the associated propagator operator. This paper reproves along the way corresponding results in almost all already known setups and also treats new ones of interest.

math.PR↗

arXiv2025

The near-critical random bond FK-percolation model

We study FK-percolation where the edge parameters are chosen as independent random variables in the near-critical regime. We show that if these parameters satisfy a natural centering condition around the critical point, then the quenched model typically exhibits critical behaviour at scales much larger than the deterministic characteristic length. More precisely, in a box of size $N$, if the homogeneous model with deterministic edge parameter $p$ looks critical in the regime $|p-p_c|\le \textrm W$, then the quenched model with random edge parameters $\mathbf p$ that typically satisfy $|\mathbf p-p_c|\le \textrm W^{1/3}$ looks critical, assuming some conjectured inequality on critical exponents, and up to logarithmic corrections. We also treat the special case of Bernoulli percolation, where we show that if one first samples non-degenerate independent random edge parameters centered around $\frac12$, and then a percolation configuration on these edges, the quenched model almost surely looks critical at large scales.

math.PR↗

arXiv2022

Magnetization in the zig-zag layered Ising model and orthogonal polynomials

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 $m\times m$ 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.

math-ph↗

arXiv2021

Conformal invariance in the quantum Ising model

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.

math.PR↗

arXiv2021

Universality of spin correlations in the Ising model on isoradial graphs

We prove universality of spin correlations in the scaling limit of the 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 $δ$ 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 \[ δ^{-\frac{1}{4}}\mathbb{E}\left[σ_{u_{1}}σ_{u_{2}}\right]\ \to\ C_σ^{2}\cdotΞ\left(|u_{1}-u_{2}|,m\right)\quad\text{as}\quadδ\to0, \] where the universal constant $C_σ$ and the function $Ξ(|u_{1}-u_{2}|,m)$ are independent of the lattice. The mass $m$ is defined by the relation $k'-1\sim 4mδ$, where $k'$ is the Baxter elliptic parameter. This includes $m$ of both signs as well as the critical case when $Ξ(r,0)=r^{-1/4}.$ 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 $z^{-1/2}$ for $m=0$ and of $z^{-1/2}e^{\pm 2m|z|}$ for $m\ne 0$. 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.

math.PR↗