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2013 Sharpness of KKL on Schreier graphs
Ryan O'Donnell, Karl Wimmer
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Electron. Commun. Probab. 18: 1-12 (2013). DOI: 10.1214/ECP.v18-1961

Abstract

Recently, the Kahn-Kalai-Linial (KKL) Theorem on influences of functions on $\{0,1\}^n$ was extended to the setting of functions on Schreier graphs. Specifically, it was shown that for an undirected Schreier graph $\text{Sch}(G,X,U)$ with log Sobolev constant $\rho$ and generating set $U$ closed under conjugation, if $f : X \to \{0,1\}$ then $$\mathcal{E}[f] \gtrsim \log(1/\text{MaxInf}[f]) \cdot \rho \cdot {\bf Var}[f].$$ Here $\mathcal{E}[f]$ denotes the average of $f$'s influences, and $\text{MaxInf}[f]$ denotes their maximum. In this work we investigate the extent to which this result is sharp. We show:

1. The condition that $U$ is closed under conjugation cannot in general be eliminated.

2. The log-Sobolev constant cannot be replaced by the modified log-Sobolev constant.

3. The result cannot be improved for the Cayley graph on $S_n$ with transpositions.

4. The result can be improved for the Cayley graph on $\mathbb{Z}_m^n$ with standard generators.

5. Talagrand's strengthened version of KKL also holds in the Schreier graph setting: $$\mathrm{avg}_{u \in U} \{\mathrm{Inf}_u[f]/\log(1/\mathrm{Inf}_u[f]) \} \gtrsim \rho \cdot {\bf Var}[f].$$

Citation

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Ryan O'Donnell. Karl Wimmer. "Sharpness of KKL on Schreier graphs." Electron. Commun. Probab. 18 1 - 12, 2013. https://doi.org/10.1214/ECP.v18-1961

Information

Accepted: 2 March 2013; Published: 2013
First available in Project Euclid: 7 June 2016

zbMATH: 06346867
MathSciNet: MR3037216
Digital Object Identifier: 10.1214/ECP.v18-1961

Subjects:
Primary: 68Q87
Secondary: 05A20, 28A99

JOURNAL ARTICLE
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