Sidorenko Inequalities for Two-Sided Group Correlation Kernels

Abstract

Sidorenko's conjecture asserts that every bipartite graph has at least the expected homomorphism density in every graph of a given edge density. Motivated by Cayley-type formulations of Sidorenko-type inequalities, we study a two-sided correlation construction on finite groups. Let $\hat{I}“$ be a finite group and let $f:\hat{I}“\to \mathbb{R}$ be a real-valued function. We define a directed kernel on $\hat{I}“$ by $${\mathcal{C}}_f(x,y)=|\hat{I}“{|}^{-1}\sum _{a_1,a_2\in \hat{I}“:x{a}_1=a_{2}y}f(a_1)f(a_2)={\mathbb{E}}_{z\in \hat{I}“}f(x^{-1}z)f(z{y}^{-1}).$$ When $f=1_A$, this is the normalized size of the intersection $xA\cap Ay$. We prove that, for every finite directed graph $F$, $$t(F,{\mathcal{C}}_f)\ge t(\overrightarrow{K_2},{\mathcal{C}}_f{)}^{e(F)}=({\mathbb{E}}_{g\in \hat{I}“}f(g){)}^{2e(F)}.$$ Equivalently, if $W_f^{\times }(x,y)=f(xy)$ is the directed product Cayley kernel on $\hat{I}“$, then the directed $1$-subdivision of every finite directed graph satisfies the same homomorphism-density lower bound in $W_f^{\times }$.