# Convergence of Ising interfaces to Schramm's SLE curves

@article{Chelkak2013ConvergenceOI, title={Convergence of Ising interfaces to Schramm's SLE curves}, author={Dmitry Chelkak and Hugo Duminil-Copin and Cl{\'e}ment Hongler and Antti Kemppainen and Stanislav Smirnov}, journal={arXiv: Mathematical Physics}, year={2013} }

## 180 Citations

### Crossing Probabilities of Multiple Ising Interfaces

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We prove that in the scaling limit, the crossing probabilities of multiple interfaces in the critical planar Ising model with alternating boundary conditions are conformally invariant expressions…

### Critical Ising interfaces in multiply-connected domains

- MathematicsProbability Theory and Related Fields
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We prove a general result on convergence of interfaces in the critical planar Ising model to conformally invariant curves absolutely continuous with respect to SLE(3). Our setup includes multiple…

### Low dimensional defects in percolation model

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We improve the Smirnov's proof of Cardy's formula for percolation on the hexagonal lattice. Then we show that the boundary trace (lamination) of the scaling limit is compatible with gluing along the…

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- PhysicsThe Annals of Probability
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We prove convergence of multiple interfaces in the critical planar q = 2 random cluster model, and provide an explicit description of the scaling limit. Remarkably, the expression for the partition…

### The Interface of the FK-representation of the Quantum Ising Model Converges to the $SLE_{16/3}$

- Physics
- 2016

We study the interface in the FK-representation of the 1D quantum Ising model and show that in the limit, it converges to the $SLE_{16/3}$ curve.

### Configurations of FK Ising interfaces and hypergeometric SLE

- Mathematics
- 2017

In this paper, we show that the interfaces in FK Ising model in any domain with 4 marked boundary points and wired--free--wired--free boundary conditions conditioned on a specific internal arc…

### A formula for boundary correlations of the critical Ising model

- Mathematics
- 2020

Given a finite rhombus tiling of a polygonal region in the plane, the associated critical Z -invariant Ising model is invariant under star-triangle transformations. We give a simple matrix formula…

### A formula for boundary correlations of the critical Ising model

- MathematicsProbability Theory and Related Fields
- 2021

Given a finite rhombus tiling of a polygonal region in the plane, the associated critical Z-invariant Ising model is invariant under star-triangle transformations. We give a simple matrix formula…

### Percolation and $O(1)$ loop model

- Mathematics
- 2021

We present an “ultimate” proof of Cardy’s formula for the critical percolation on the hexagonal lattice [23], showing the existence of the universal and conformally invariant scaling limit of…

### Conformal invariance in the FK-representation of the quantum Ising model and convergence of the interface to the $$\mathrm{SLE}_{16/3}$$SLE16/3

- Mathematics
- 2019

We study the interface in the FK-representation of the 1D quantum Ising model and show that in the limit, it converges to the $$\mathrm{SLE}_{16/3}$$SLE16/3 curve.

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