# The incipient infinite cluster in two-dimensional percolation

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## Summary

Let*P*_{ p } be the probability measure on the configurations of occupied and vacant vertices of a two-dimensional graphG, under which all vertices are independently occupied (respectively vacant) with probability*p* (respectively 1-*p*). Let*P*_{ H } be the critical probability for this system and*W* the occupied cluster of some fixed vertex*w*_{0}. We show that for many graphsG, such as\(\mathbb{Z}^2 \), or its covering graph (which corresponds to bond percolation on\(\mathbb{Z}^2 \)), the following two conditional probability measures converge and have the same limit,*v* say:

- i)
*P*_{ pH }{·∣*w*_{0}is connected by an occupied path to the boundary of the square [-*n,n*]^{2}} as*n*→∞, - ii)
*P*_{ p }{·∣*W*is infinite} as*p*↓*p*_{ H }.

On a set of*v*-measure one,*w*_{0} belongs to a unique infinite occupied cluster,*W*W} say. We propose that*W*W} be used for the “incipient infinite cluster”. Some properties of the density of*W*W} and its “backbone” are derived.

## Keywords

Stochastic Process Probability Measure Probability Theory Conditional Probability Mathematical Biology## Preview

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## References

- 1.Alexander, S., Orbach, R.: Density of states on fractals: «fractons», J. Physique Lett.
**43**, L 625–631 (1982)Google Scholar - 2.Harris, T.E.: A lower bound for the critical probability in a certain percolation process. Proc. Cambr. Phil. Soc.
**56**, 13–20 (1960)Google Scholar - 3.Hopf, E.: An inequality for positive linear integral operators, J. Math. Mech.
**12**, 683–692 (1963)Google Scholar - 4.Karlin, S., Studden, W.J.: Tchebycheff systems: with applications in analysis and statistics. New York: Interscience Publ. (1966)Google Scholar
- 5.Kesten, H.: The critical probability of bond percolation on the square lattice equals 1/2. Comm. Math. Phys.
**74**, 41–59 (1980)Google Scholar - 6.Kesten, H.: Percolation theory for mathematicians. Boston: Birkhäuser (1982)Google Scholar
- 7.Kesten, H.: Subdiffusive behavior of random walk on random clusters. To appear in Ann. Inst. H. Poincaré (1987)Google Scholar
- 8.Leyvraz, F., Stanley, H.E.: To what class of fractals does the Alexander-Orbach conjecture apply? Phys. Rev. Lett.
**51**, 2048–2051 (1983)Google Scholar - 9.Neveu, J.: Mathematical foundations of the calculus of probability. Holden-Day (1965)Google Scholar
- 10.Russo, L.: A note on percolation. Z. Wahrscheinlichkeitstheor. Verw. Geb.
**43**, 39–48 (1978)Google Scholar - 11.Russo, L.: On the critical percolation probabilities. Z. Wahrscheinlichkeitstheor. Verw. Geb.
**56**, 229–237 (1981)Google Scholar - 12.Seymour, P.D., Welsh, D.J.A.: Percolation probabilities on the square lattice. Ann. Discrete Math.
**3**, 227–245 (1978)Google Scholar - 13.Smythe, R.T., Wierman, J.C.: First-passage percolation on the square lattice, Lecture Notes in Mathematics
**671**. Berlin, Heidelberg, New York: Springer (1978)Google Scholar - 14.Stanley, H.E., Coniglio, A.: Fractal structure of the incipient infinite cluster in percolation. In: Percolation Structures and Processes, vol. 5, pp. 101–120. Ann. Israel Phys. Soc. (1983)Google Scholar
- 15.van den Berg, J., Kesten, H.: Inequalities with applications to percolation and reliability. J. Appl. Probab.
**22**, 556–569 (1985)Google Scholar - 16.Wierman, J.C.: Bond percolation on honeycomb and triangular lattices. Adv. Appl. Prob.
**13**, 293–313 (1981)Google Scholar