Abstract
In this chapter we consider the subcritical phase of bond percolation on Ld when d ≥ 2; that is, we suppose that the edge-probability p satisfies p < p c . In this phase, all open clusters are finite almost surely and furthermore have finite mean size. We are interested in such quantities as (i) estimates for the probability of an open path joining two vertices x and y when the distance between x and y is large, and (ii) estimates for the rate of decay of P p (|C| = n) as n → ∞. Such estimates contain information about the structure of the process over long ranges, and as applications of such estimates we shall prove that x(p) and κ(p) are analytic functions of p when p < p c .
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Notes
Hammersley, J. M. 1957a Percolation processes. Lower bounds for the critical probability, Annals of Mathematical Statistics 28, 790–795.
Kesten, H. 1985 First-passage percolation and a higher dimensional generalization, in Particle Systems, Random Media and Large Deviations, ed. R. Durrett, 235–251, Contemporary Mathematics 41, American Mathematical Society, Providence, Rhode Island.
McDiarmid, C. J. H. 1981 General percolation and random graphs Advances in Applied Probability 13, 40–60.
Kesten, H. 1987e Percolation theory and first-passage percolation, Annals of Probability 15, 1231–1271.
Grimmett, G. R. 1979 Central limit theorems in percolation theory, unpublished manuscript. 1981a Critical sponge dimensions in percolation theory, Advances in Applied Probability 13, 314–324.
Grigorchuk, R. I. 1983 On Milnor’s problem of group growth, Soviet Mathematics Doklady 28, 23–26.
Chayes, J. T. and Chayes, L. 1986a Percolation and random media, in Critical Phenomena, Random Systems and Gauge Theories, Les Houches, Session XLIII, 1984, eds. K. Osterwalder and R. Stora, 1001–1142, Elsevier, Amsterdam.
Chayes, J. T. and Chayes, L. 1986b Critical points and intermediate phases on wedges of V, Journal of Physics A: Mathematical and General 19, 3033–3048.
Cox, J. T. and Grimmett, G. R. 1984 Central limit theorems for associated random variables and the percolation model, Annals of Probability 12, 514–528.
Durrett, R. 1985a Some general results concerning the critical exponents of percolation processes, Zeitschrift für Wahrscheinlichkeitstheorie and Verwandte Gebiete 69, 421–437.
Chandler, R., Koplick, J., Lerman, K., and Willemsen, J. F. 1982 Capillary displacement and percolation in porous media, Journal of Fluid Mechanics 119, 249–267.
Chayes, J. T. and Chayes, L. 1986a Percolation and random media, in Critical Phenomena, Random Systems and Gauge Theories, Les Houches, Session XLIII, 1984, eds. K. Osterwalder and R. Stora, 1001–1142, Elsevier, Amsterdam.
Kesten, H. 1980b On the time constant and path length of first passage percolation, Advances in Applied Probability 12, 848–863.
Kunz, H. and Souillard, B. 1978 Essential singularity in percolation problems and asymptotic behavior of cluster size distribution, Journal of Statistical Physics 19, 77–106.
Neaderhouser, C. C. 1978 Limit theorems for multiply indexed mixing random variables with application to Gibbs random fields, Annals of Probability 6, 207–215.
Wierman, J. C.1985b Critical percolation probabilities, in Random Graphs ‘83, ed. M. Karonski, 349–359, Annals of Discrete Mathematics 28, North-Holland, Amsterdam.
Tasaki, H. 1987a Geometric critical exponent inequalities for general random cluster models, Journal of Statistical Physics 49, 841–847.
Smythe, R. T. and Wierman, J. C. 1978 First-Passage Percolation on the Square Lattice, Lecture Notes in Mathematics no. 671, Springer, Berlin.
Newman, C. M. and Schulman, L. S. 1981b Number and density of percolating clusters, Journal of Physics A: Mathematical and General 14, 1735–1743.
Kesten, H. 1981 Analyticity properties and power law estimates in percolation theory, Journal of Statistical Physics 25, 717–756.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 1989 Springer Science+Business Media New York
About this chapter
Cite this chapter
Grimmett, G. (1989). The Subcritical Phase. In: Percolation. Springer, New York, NY. https://doi.org/10.1007/978-1-4757-4208-4_5
Download citation
DOI: https://doi.org/10.1007/978-1-4757-4208-4_5
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4757-4210-7
Online ISBN: 978-1-4757-4208-4
eBook Packages: Springer Book Archive