| Título: | Sharpness of KKL on Schreier graphs |
| Autores: |
O'Donnell, Ryan; Carnegie Mellon University Wimmer, Karl; Duquesne University |
| Fecha: | 2013-01-03 |
| Publicador: | Electronic communications in probability |
| Fuente: |
 |
| Tipo: | Peer-reviewed Article |
| Tema: |
boolean functions; KKL; Cayley graphs; Schreier graphs; log-sobolev constant; Orlicz norms 68Q87; 28A99; 05A20 |
| Descripción: | 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: $$\mathrm{avg}_{u \in U} \{\mathrm{Inf}_u[f]/\log(1/\mathrm{Inf}_u[f]) \} \gtrsim \rho \cdot {\bf Var}[f].$$ |
| Idioma: | Inglés |
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