# Connection probabilities and RSW-type bounds for the FK Ising model

@article{DuminilCopin2009ConnectionPA, title={Connection probabilities and RSW-type bounds for the FK Ising model}, author={Hugo Duminil-Copin and Cl{\'e}ment Hongler and Pierre Nolin}, journal={arXiv: Probability}, year={2009} }

We prove Russo-Seymour-Welsh-type uniform bounds on crossing probabilities for the FK Ising model at criticality, independent of the boundary conditions. Our proof relies mainly on Smirnov's fermionic observable for the FK Ising model, which allows us to get precise estimates on boundary connection probabilities. It remains purely discrete, in particular we do not make use of any continuum limit, and it can be used to derive directly several noteworthy results - some new and some not - among…

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## References

SHOWING 1-10 OF 12 REFERENCES

### Universality in the 2D Ising model and conformal invariance of fermionic observables

- Mathematics, Physics
- 2009

It is widely believed that the celebrated 2D Ising model at criticality has a universal and conformally invariant scaling limit, which is used in deriving many of its properties. However, no…

### On the random-cluster model: I. Introduction and relation to other models

- Mathematics, Computer Science
- 1972

### Generalization of the Fortuin-Kasteleyn-Swendsen-Wang representation and Monte Carlo algorithm.

- Computer SciencePhysical review. D, Particles and fields
- 1988

A simple explanation of the Swendsen-Wang algorithm for Potts models in terms of a joint model of Potts spin variables interacting with bond occupation variables is given and how to generalize this representation to arbitrary models is shown.

### Crystal statistics. I. A two-dimensional model with an order-disorder transition

- Materials Science
- 1944

The partition function of a two-dimensional "ferromagnetic" with scalar "spins" (Ising model) is computed rigorously for the case of vanishing field. The eigenwert problem involved in the…

### , S aling relations for 2 D - per olation

- Comm . Math . Phys .
- 1987

### Criti al exponents for two - dimensional per ola - tion

- Math . Res . Lett .
- 2006

### In uen e and sharp - threshold theoremsfor monotoni measures

- Ann . Probab .
- 2006

### In nite onformalsymmetry in two - dimensional quantum eld theory

- 1984

### Statisti s of the two - dimensional ferromagnet , I , II

- Phys . Rev .
- 1941