39 research outputs found
Conformal Invariance of Spin Correlations in the Planar Ising Model
We rigorously prove the existence and the conformal invariance of scaling
limits of the magnetization and multi-point spin correlations in the critical
Ising model on arbitrary simply connected planar domains. This solves a number
of conjectures coming from the physical and the mathematical literature. The
proof relies on convergence results for discrete holomorphic spinor observables
and probabilistic techniques.Comment: Changes in this version: the explicit formula for n-point spin
correlations is proved in full generality. The appendix is rewritten
completely and contains this new proof, the introduction is changed
accordingly, the presentation in Sections 2.5 and 2.7(2.8) is rearranged
slightly. 42 pages, 2 figure
Magnetization in the zig-zag layered Ising model and orthogonal polynomials
We discuss the magnetization in the -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 via 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 as a Toeplitz+Hankel determinant for the
homogeneous sub-critical model in presence of a boundary magnetic field.Comment: minor updates + Section 5.3 added; 38 page
Conformal invariance of crossing probabilities for the Ising model with free boundary conditions
We prove that crossing probabilities for the critical planar Ising model with
free boundary conditions are conformally invariant in the scaling limit, a
phenomenon first investigated numerically by Langlands, Lewis and Saint-Aubin.
We do so by establishing the convergence of certain exploration processes
towards SLE. We also construct an exploration tree
for free boundary conditions, analogous to the one introduced by Sheffield.Comment: 18 pages, 4 figures, v2: journal versio