Electronic Communications in Probability

Sharpness of KKL on Schreier graphs

Ryan O'Donnell and Karl Wimmer

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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: $$\mathrm{avg}_{u \in U} \{\mathrm{Inf}_u[f]/\log(1/\mathrm{Inf}_u[f]) \} \gtrsim \rho \cdot {\bf Var}[f].$$

Article information

Source
Electron. Commun. Probab., Volume 18 (2013), paper no. 18, 12 pp.

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

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1465315557

Digital Object Identifier
doi:10.1214/ECP.v18-1961

Mathematical Reviews number (MathSciNet)
MR3037216

Zentralblatt MATH identifier
06346867

Subjects
Primary: 68Q87: Probability in computer science (algorithm analysis, random structures, phase transitions, etc.) [See also 68W20, 68W40]
Secondary: 28A99: None of the above, but in this section 05A20: Combinatorial inequalities

Keywords
boolean functions KKL Cayley graphs Schreier graphs log-sobolev constant Orlicz norms

Rights
This work is licensed under a Creative Commons Attribution 3.0 License.

Citation

O'Donnell, Ryan; Wimmer, Karl. Sharpness of KKL on Schreier graphs. Electron. Commun. Probab. 18 (2013), paper no. 18, 12 pp. doi:10.1214/ECP.v18-1961. https://projecteuclid.org/euclid.ecp/1465315557


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