The Match Set of a Random Permutation Has the FKG Property

Abstract

We prove a conjecture of Joag-Dev and Goel that if $M = M(\sigma) = \{i: \sigma(i) = i\}$ is the (random) match set, or set of fixed points, of a random permutation $\sigma$ of $1,2,\ldots, n$, then $f(M)$ and $g(M)$ are positively correlated whenever $f$ and $g$ are increasing real-valued set functions on $2^{\{1,\ldots, n\}}$, i.e., $Ef(M)g(M) \geq Ef(M)Eg(M)$. No simple use of the FKG or Ahlswede-Daykin inequality seems to establish this, despite the fact that the FKG hypothesis is "almost" satisfied. Instead we reduce to the case where $f$ and $g$ take values in $\{0, 1\}$, and make a case-by-case argument: Depending on the specific form of $f$ and $g$, we move the probability weights around so as to make them satisfy the FKG or Ahlswede-Daykin hypotheses, without disturbing the expectations $Ef, Eg, Efg$. This approach extends the methodology by which FKG-style correlation inequalities can be proved.