On the rate of convergence for critical crossing probabilities

Binder, I.; Chayes, L.; Lei, H. K.

Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015) no. 2, pp. 672-715.

Dans le modèle de percolation sur le réseau triangulaire et pour certaines généralisations pour lesquelles la formule de Cardy a été établie, nous démontrons un taux de convergence en loi de puissance des probabilités de percolation vers la formule de Cardy.

For the site percolation model on the triangular lattice and certain generalizations for which Cardy’s Formula has been established we acquire a power law estimate for the rate of convergence of the crossing probabilities to Cardy’s Formula.

[1] M. Aizenman, J. T. Chayes, L. Chayes, J. Fröhlich and L. Russo. On a sharp transition from area law to perimeter law in a system of random surfaces. Comm. Math. Phys. 92 (1983) 19–69. | DOI | MR | Zbl

[2] V. Beffara. Cardy’s Formula on the triangular lattice, the easy way. In Universality and Renormalization 39–45. Fields Institute Communications 50. Amer. Math. Soc., Providence, RI, 2007. | MR | Zbl

[3] C. Beneš, F. Johansson Viklund and M. J. Kozdron. On the rate of convergence of loop-erased random walk to ${\mathrm{SLE}}_2$. Comm. Math. Phys. 318 (2013) 307–354. | DOI | MR | Zbl

[4] I. Binder, L. Chayes and H. K. Lei. On convergence to ${\mathrm{SLE}}_6$ I: Conformal invariance for certain models of the bond-triangular type. J. Stat. Phys. 141 (2) (2010) 359–390. | MR | Zbl

[5] I. Binder, L. Chayes and H. K. Lei. On convergence to ${\mathrm{SLE}}_6$ II: Discrete approximations and extraction of Cardy’s formula for general domains. J. Stat. Phys. 141 (2) (2010) 391–408. | MR | Zbl

[6] F. Camia and C. M. Newman. Critical percolation exploration path and ${\mathrm{SLE}}_6$: A proof of convergence. Probab. Theory Related Fields 139 (2007) 473–519. | DOI | MR | Zbl

[7] J. L. Cardy. Critical percolation in finite geometries. J. Phys. A 25 (1992) 201–206. | DOI | MR | Zbl

[8] L. Chayes. Discontinuity of the spin-wave stiffness in the two-dimensional $XY$ model. Comm. Math. Phys. 197 (1998) 623–640. | DOI | MR | Zbl

[9] L. Chayes. Mean field analysis of low-dimensional systems. Comm. Math. Phys. 292 (2009) 303–341. | DOI | MR | Zbl

[10] L. Chayes and H. K. Lei. Cardy’s formula for certain models of the bond-triangular type. Rev. Math. Phys. 19 (5) (2007) 511–565. | MR | Zbl

[11] T. E. Harris. A lower bound for the critical probability in a certain percolation process. Proc. Cambridge Philos. Soc. 56 (1960) 13–20. | DOI | MR | Zbl

[12] G. F. Lawler. Conformally Invariant Processes in the Plane. Mathematical Surveys and Monographs 114. Amer. Math. Soc., Providence, RI, 2005. | MR | Zbl

[13] D. Mendelson, A. Nachmias and S. S. Watson. Rate of convergence for Cardy’s formula. Comm. Math. Phys. 329 (1) (2014) 29–56. | MR | Zbl

[14] S. Smirnov. Critical percolation in the plane: Conformal invariance, Cardy’s formula, scaling limits. C. R. Acad. Sci. Paris Ser. I Math. 333 (2001) 239–244. | MR | Zbl

[15] F. J. Viklund. Convergence rates for loop-erased random walk and other loewner curves. Ann. Probab. To appear, 2015. Available at http://arxiv.org/abs/1205.5734. | MR | Zbl

[16] S. E. Warschawski. On the degree of variation in conformal mapping of variable regions. Trans. Amer. Math. Soc. 69 (2) (1950) 335–356. | MR | Zbl

[17] W. Werner. Lectures on two-dimensional critical percolation. In Statistical Mechanics 297–360. IAS/Park City Math. Ser. 16. Amer. Math. Soc., Providence, RI, 2009. | MR | Zbl