Influence and sharp-threshold theorems for monotonic measures

By B. Graham, Geoffrey Grimmett. 2005

66 citations recorded by the source corpus.

The influence theorem for product measures on the discrete space (0, 1} N may be extended to probability measures with the property of monotonicity (which is equivalent to "strong positive association"). Corresponding results are valid for probability measures on the cube [0, 1] N that are absolutely continuous with respect to Lebesgue measure. These results lead to a sharp-threshold theorem for measures of random-cluster type, and this may be applied to box crossings in the two-dimensional random-cluster model.

Subfields: Management Science and Operations Research, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/5f31602364e23c94b827f96dd54c972acba7ce1b

Citations in the corpus

• Sharp phase transition for the random-cluster and Potts models via decision trees (2017, unknown): “This result was originally proved in [BD12], where exponential decay of correlations below pc is proved using Russo-Seymour-Welsh type arguments and a generalization [GG06] of the KKL result [KKL88, BKK92].”
• The self-dual point of the two-dimensional random-cluster model is critical for q ≥ 1 (2010, unknown): “The proofs are directly extracted from the Grimmett’s monograph [18].”