Strict monotonicity for critical points in percolation and ferromagnetic models

Michael Aizenman, Geoffrey Grimmett

Research output: Contribution to journalArticlepeer-review

87 Scopus citations


When is the numerical value of the critical point changed by an enhancement of the process or of the interaction? Ferromagnetic spin models, independent percolation, and the contact process are known to be endowed with monotonicity properties in that certain enhancements are capable of shifting the corresponding phase transition in only an obvious direction, e. g., the addition of ferromagnetic couplings can only increase the transition temperature. The question explored here is whether enhancements do indeed change the value of the critical point. We present a generally applicable approach to this issue. For ferromagnetic Ising spin systems, with pair interactions of finite range in d≥2 dimensions, it is shown that the critical temperature Tc is strictly monotone increasing in each coupling, with the first-order derivatives bounded by positive functions which are continuous on the set of fully d-dimensional interactions. For independent percolation, with 0<pc<1, we prove that any "essential enhancement" of the process has an effect on the critical probability, a result with applications to the question of the existence of "entanglements" and to invasion percolation with trapping.

Original languageEnglish (US)
Pages (from-to)817-835
Number of pages19
JournalJournal of Statistical Physics
Issue number5-6
StatePublished - Jun 1 1991

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics


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