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 $Γ$ be a finite group and let $f:Γ\to\mathbb{R}$ be a real-valued function. We define a directed kernel on $Γ$ by $$\mathcal C_f(x,y)=|Γ|^{-1}\sum_{a_1,a_2\inΓ:\, xa_1=a_2y} f(a_1)f(a_2)=\mathbb{E}_{z\inΓ} f(x^{-1}z)f(zy^{-1}).$$ When $f=\mathbf{1}_A$, this is the normalized size of the intersection $xA\cap Ay$. We prove that, for every finite directed graph $F$, $$t(F,\mathcal C_f)\geq t(\overrightarrow{K_2},\mathcal C_f)^{e(F)}=(\mathbb{E}_{g\inΓ}f(g))^{2e(F)}.$$ Equivalently, if $W_f^\times(x,y)=f(xy)$ is the directed product Cayley kernel on $Γ$, then the directed $1$-subdivision of every finite directed graph satisfies the same homomorphism-density lower bound in $W_f^\times$.