Lower bound on the correlation between monotone families in the average case
Correlation inequalities between monotone functions play an important role in numerous areas, including probability, combinatorics etc. This paper considers monotone functions on the discrete cube \(\{0,1\}^n\) endowed with the uniform measure, and especially Boolean functions that can be treated as characteristic functions of subsets of the discrete cube. It is shown that when the correlation is averaged over all pairs \(A,B \in T\), for any family \(T\) of monotone subsets of \(\{0,1\}^n\), the lower bound asserted in [\textit{M. Talagrand}, Combinatorica 16, No. 2, 243--258 (1996; Zbl 0861.05008)] can be improved and more precise estimates on the average correlation can be given.
scientific article; zbMATH DE number 5564307