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Sharpness of KKL on Schreier graphs


 
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1. Title Title of document Sharpness of KKL on Schreier graphs
 
2. Creator Author's name, affiliation, country Ryan O'Donnell; Carnegie Mellon University; United States
 
2. Creator Author's name, affiliation, country Karl Wimmer; Duquesne University; United States
 
3. Subject Discipline(s)
 
3. Subject Keyword(s) boolean functions; KKL; Cayley graphs; Schreier graphs; log-sobolev constant; Orlicz norms
 
3. Subject Subject classification 68Q87; 28A99; 05A20
 
4. Description 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: $$\mathrm{avg}_{u \in U} \{\mathrm{Inf}_u[f]/\log(1/\mathrm{Inf}_u[f]) \} \gtrsim \rho \cdot {\bf Var}[f].$$
 
5. Publisher Organizing agency, location
 
6. Contributor Sponsor(s) NSF, BSF, Sloan Foundation
 
7. Date (YYYY-MM-DD) 2013-03-02
 
8. Type Status & genre Peer-reviewed Article
 
8. Type Type
 
9. Format File format PDF
 
10. Identifier Uniform Resource Identifier http://ecp.ejpecp.org/article/view/1961
 
10. Identifier Digital Object Identifier 10.1214/ECP.v18-1961
 
11. Source Journal/conference title; vol., no. (year) Electronic Communications in Probability; Vol 18
 
12. Language English=en en
 
14. Coverage Geo-spatial location, chronological period, research sample (gender, age, etc.)
 
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