Abstract
We prove a relation between the radius and volume of a two-dimensional percolation cluster. This implies for 2D percolation that the critical exponents δ and η satisfy η = 4/(δ + 1) (provided η exists).
Research supported by the NSF through a grant to Cornell University.
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H. Kesten, The incipient infinite cluster in two-dimensional percolation, submitted to Theor. Probab. Rel. Fields.
H. Kesten, Subdiffusive behavior of random walk on a random cluster, submitted to Ann. Inst. H. Poincaré.
J. van den Berg and H. Kesten, Inequalities with applications to percolation and reliability, J. Appl. Prob. 22, 556–569 (1985).
M.J. Buckingham and J.D. Gunton, Correlations at the critical point of the Ising model, Phys. Rev. 178, 848–853 (1969).
M.E. Fisher, Rigorous inequalities for critical-point correlation exponents, Phys. Rev. 180, 594–600 (1969).
G. Stell, Extension of the Ornstein-Zernike theory of the critical region, Phys. Rev. Lett. 20, 533–536 (1968).
M.E. Fisher, The theory of equilibrium statistical phenomenon, Rep. Prog. Phys. 30, 615–730 (1967).
D. Stauffer, Scaling properties of percolation clusters, pp. 9–25 in Disordered Systems and Localization, C. Castellani, C. Di Castro and L. Peliti eds., Lecture notes in physics, vol. 149, Springer, 1981.
P.A. Seymour and D.J.A. Welsh, Percolation probabilities on the square lattice, Ann. Discrete Math. 3, 227–245 (1978).
L. Russo, On the critical percolation probabilities, Z. Wahrsch. verw. Geb. 56, 229–237 (1981).
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© 1987 Springer-Verlag New York, Inc.
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Kesten, H. (1987). A Scaling relation at criticality for 2D-Percolation. In: Kesten, H. (eds) Percolation Theory and Ergodic Theory of Infinite Particle Systems. The IMA Volumes in Mathematics and Its Applications, vol 8. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-8734-3_12
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DOI: https://doi.org/10.1007/978-1-4613-8734-3_12
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