Critical exponents for two-dimensional percolation. (English) Zbl 1009.60087

Smirnov, Stanislav; Werner, Wendelin

Math. Res. Lett. 8, No. 5-6, 729-744 (2001).

The paper deals with problems of the existence and values of critical exponents for site percolation on the triangular lattice. Each vertex of a triangular lattice is open with probability \(p\) (and closed with probability \(1-p\)). Then \(p={1\over 2}\) is the critical value. The authors give, besides other results, asymptotic formulas (for \(p\to{1\over 2}\)) for the probability \(\theta(p)\) that the origin belongs to an infinite cluster of open vertices, of the average cardinality \(\chi(p)\) of finite clusters, and of the correlation length \(\xi(p)\) corresponding to the typical radius of a finite cluster. In particular, it holds \[ \begin{aligned} \theta(p)&= (p-1/2)^{5/36+o(1)} \quad \text{for} \;p\to 1/2,\\ \chi(p)&= (p-1/2)^{-43/18+o(1)} \quad \text{for} \;p\to 1/2,\\ \xi(p)&= (p-1/2)^{-4/3+o(1)} \quad \text{for} \;p\to 1/2. \end{aligned} \] The survey paper contains also many other results concerning the subject.

Reviewer: Martin Janžura (Praha)

MSC:

60K35	Interacting random processes; statistical mechanics type models; percolation theory
82B26	Phase transitions (general) in equilibrium statistical mechanics
82B43	Percolation

Keywords:

site percolation; triangular lattice; critical exponent