• Corpus ID: 119329824

# Another proof of $p_c = \frac{\sqrt{q}}{1+\sqrt{q}}$ on $\mathbb{Z}^2$ with $q \in [1,4]$ for random-cluster model

@article{Mukoseeva2018AnotherPO,
title={Another proof of \$p\_c = \frac\{\sqrt\{q\}\}\{1+\sqrt\{q\}\}\$ on \$\mathbb\{Z\}^2\$ with \$q \in [1,4]\$ for random-cluster model},
author={Ekaterina Mukoseeva and Daria Smirnova},
journal={arXiv: Mathematical Physics},
year={2018}
}
• Published 9 December 2018
• Mathematics
• arXiv: Mathematical Physics
In this paper we give another proof of $p_c = \frac{\sqrt{q}}{1+\sqrt{q}}$ on $\mathbb{Z}^2$ with $q \in [1,4]$, based on the method of parafermionic observables.

## References

SHOWING 1-10 OF 18 REFERENCES
Connective constant for a weighted self-avoiding walk on $\mathbb{Z}^2$
We consider a self-avoiding walk on the dual $\mathbb{Z}^2$ lattice. This walk can traverse the same square twice but cannot cross the same edge more than once. The weight of each square visited by
The self-dual point of the two-dimensional random-cluster model is critical for q ≥ 1
• Mathematics
• 2010
We prove a long-standing conjecture on random-cluster models, namely that the critical point for such models with parameter q ≥ 1 on the square lattice is equal to the self-dual point {p_{sd}(q) =
Macroscopic loops in the loop $O(n)$ model at Nienhuis' critical point
• Mathematics
Journal of the European Mathematical Society
• 2020
The loop $O(n)$ model is a model for a random collection of non-intersecting loops on the hexagonal lattice, which is believed to be in the same universality class as the spin $O(n)$ model. It has
A New Proof of the Sharpness of the Phase Transition for Bernoulli Percolation and the Ising Model
• Mathematics, Computer Science
• 2015
A new proof of the sharpness of the phase transition for Bernoulli percolation and the Ising model is provided and it is proved that the two-point correlation functions decay exponentially fast in the distance for β < βc.
On the critical parameters of the $q\ge4$ random-cluster model on isoradial graphs
• Mathematics
• 2015
The critical surface for random-cluster model with cluster-weight $q\ge 4$ on isoradial graphs is identified using parafermionic observables. Correlations are also shown to decay exponentially fast
Continuity of the phase transition for planar random-cluster and Potts models with 1 ≤ q ≤ 4
• Mathematics
• 2015
This article studies the planar Potts model and its random-cluster representation. We show that the phase transition of the nearest-neighbor ferromagnetic q-state Potts model on Z 2 is continuous for
The phase transitions of the planar random-cluster and Potts models with q larger than 1 are sharp
• Mathematics
• 2014
We prove that random-cluster models with q larger than 1 on a variety of planar lattices have a sharp phase transition, that is that there exists some parameter p_c below which the model exhibits
Percolation ?
• Mathematics
• 1982
572 NOTICES OF THE AMS VOLUME 53, NUMBER 5 Percolation is a simple probabilistic model which exhibits a phase transition (as we explain below). The simplest version takes place on Z2, which we view