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July, 1988 The Match Set of a Random Permutation Has the FKG Property
Peter C. Fishburn, Peter G. Doyle, L. A. Shepp
Ann. Probab. 16(3): 1194-1214 (July, 1988). DOI: 10.1214/aop/1176991685


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.


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Peter C. Fishburn. Peter G. Doyle. L. A. Shepp. "The Match Set of a Random Permutation Has the FKG Property." Ann. Probab. 16 (3) 1194 - 1214, July, 1988.


Published: July, 1988
First available in Project Euclid: 19 April 2007

zbMATH: 0649.60006
MathSciNet: MR942763
Digital Object Identifier: 10.1214/aop/1176991685

Primary: 60B15
Secondary: 06A10 , 06D99 , 60E15

Keywords: Ahlswede-Daykin inequality , correlated functions , Fixed points , FKG inequality , Random permutations

Rights: Copyright © 1988 Institute of Mathematical Statistics


Vol.16 • No. 3 • July, 1988
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