Percolation pp 82-108 | Cite as

The Subcritical Phase

  • Geoffrey Grimmett


In this chapter we consider the subcritical phase of bond percolation on L d 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 .


Correlation Length Incident Edge Open Cluster Percolation Process Connectivity Function 
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Copyright information

© Springer Science+Business Media New York 1989

Authors and Affiliations

  • Geoffrey Grimmett
    • 1
  1. 1.School of MathematicsUniversity of BristolBristolEngland

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