On the rate of convergence for critical crossing probabilities. (English. French summary) Zbl 1330.82025

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

Ann. Inst. Henri Poincaré, Probab. Stat. 51, No. 2, 672-715 (2015).

Starting with [S. Smirnov, C. R. Acad. Sci., Paris, Sér. I, Math. 333, No. 3, 239–244 (2001; Zbl 0985.60090)] and continuing up to [W. Werner, in: Statistical mechanics. Papers based on the presentations at the IAS/PCMI summer conference, Park City, UT, USA, July 1–21, 2007. Providence, RI: American Mathematical Society (AMS); Princeton, NJ: Institute for Advanced Study. 297–358 (2009; Zbl 1180.82003)], the validity of Cardy’s formula, which describes the limit of crosssing probabilities for certain percolation models (and the subsequent validity of an \(SLE_6\) description for an associated limiting explorer process) has been well established. The mechanism of how Cardy’s formula comes into play has become purely understood. The purpose of the present paper is an investigation of the rate of convergence of crossing probabilities to the Cardy’s formula, viewed as the percolation observable. A power law estimate has been established for the site percolation model on the triangular lattice and certain generalizations, for which Cardy’s formula has been so far established. That holds true for the rate of convergence in any domain with boundary dimension less than two. The procedure invented here is akin to the approach of T. E. Harris [Proc. Camb. Philos. Soc. 56, 13–20 (1960; Zbl 0122.36403)], in his study of the critical state at a time when detailed information about the nature of the state was unavailable.

Reviewer: Piotr Garbaczewski (Opole)

MSC:

82B43	Percolation
60K35	Interacting random processes; statistical mechanics type models; percolation theory
82B27	Critical phenomena in equilibrium statistical mechanics
82B20	Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics

Keywords:

critical percolation in the plane; crossing probability; triangular lattice; conformal invariance; Cardy’s formula; scaling limits; Harris systems; holomorphicity

Citations:

Zbl 0985.60090; Zbl 1180.82003; Zbl 0122.36403

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