Sidorenko Inequalities for Two-Sided Group Correlation Kernels

Yuqi Zhao

arxiv: 2606.23018 · v1 · pith:PP5YXECOnew · submitted 2026-06-22 · 🧮 math.CO

Sidorenko Inequalities for Two-Sided Group Correlation Kernels

Yuqi Zhao This is my paper

Pith reviewed 2026-06-26 08:16 UTC · model grok-4.3

classification 🧮 math.CO

keywords Sidorenko conjecturetwo-sided correlation kernelshomomorphism densitiesdirected graphsCayley kernelsfinite groupsproduct kernels

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The pith

For the two-sided correlation kernel C_f on a finite group, the homomorphism density t(F, C_f) is at least the square of the average of f raised to twice the number of edges, for every finite directed graph F.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces a directed kernel C_f on a finite group Γ that averages the product f(a1)f(a2) over pairs where xa1 equals a2 y, equivalently expressed as an expectation over z of f(x inverse z) times f(zy inverse). It proves that this kernel obeys a Sidorenko-type lower bound: the density of any finite directed graph F inside C_f is at least the density of one directed edge raised to the power of the number of edges in F, which simplifies to the average of f over the group raised to twice the edge count. The same bound holds when the kernel is rewritten as the product Cayley kernel W_f^times on the directed 1-subdivision of F. A reader cares because the result supplies an explicit infinite family of kernels, built from arbitrary real functions on groups, where the inequality is known to hold.

Core claim

The central claim is that for every finite directed graph F and every real-valued f on a finite group Γ, the homomorphism density satisfies t(F, C_f) ≥ t(→K_2, C_f)^{e(F)} = (E_{g in Γ} f(g))^{2 e(F)}, where C_f is the two-sided correlation kernel given by the normalized sum over a1, a2 with xa1 = a2 y of f(a1)f(a2). Equivalently the same lower bound holds for the product kernel W_f^times(x,y) = f(xy) on the directed 1-subdivision of F.

What carries the argument

The two-sided correlation kernel C_f, which records the normalized size of intersections xA ∩ A y when f is an indicator and carries the lower bound through the group multiplication structure.

If this is right

The directed 1-subdivision of every finite directed graph satisfies the same homomorphism-density lower bound inside the product Cayley kernel W_f^times.
When f is the indicator function of a subset A the kernel C_f counts the normalized measure of intersections xA ∩ Ay.
The inequality supplies an explicit family of directed kernels on which Sidorenko-type statements are verified for all directed graphs F.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The construction may extend to continuous groups by replacing sums with integrals, though the paper does not address this case.
The bound could be used to certify Sidorenko-type inequalities for kernels that arise as limits of these group-based examples.
Testing the inequality on small non-abelian groups and specific f would give concrete numerical checks of the equality case.

Load-bearing premise

The result assumes the group Γ is finite so that the normalized average over group elements is well-defined and the two expressions for C_f can be equated.

What would settle it

A finite group Γ, a function f on Γ, and a directed graph F for which the measured homomorphism density t(F, C_f) falls below (average of f)^{2e(F)} would falsify the claim.

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

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper defines a two-sided group correlation kernel and proves the Sidorenko lower bound holds for it on every directed graph.

read the letter

The main takeaway is that this paper constructs an explicit kernel C_f on a finite group via the two-sided average E_z f(x^{-1}z) f(zy^{-1}) and shows t(F, C_f) is at least (E f)^{2e(F)} for any directed F.

What is new is the two-sided formulation itself, which does not collapse to the one-sided Cayley kernels in the motivation section. The proof links the bound to the product kernel W_f^times on 1-subdivisions of F, which are bipartite, so the result amounts to verifying the inequality algebraically for those specific graphs.

The construction works cleanly for arbitrary real f and uses only the bijectivity of group multiplications, which checks out on the finite-group hypotheses. The equivalence to the subdivision case is a useful observation.

The soft spot is that the result stays inside this one family of kernels and does not address the general Sidorenko conjecture or produce new cases for arbitrary graphs. The finiteness assumption is necessary and stated, but it limits scope. No circularity appears in the claim.

This is for people working on algebraic or group-based methods in extremal combinatorics. A reader already looking at Sidorenko variants or correlation kernels would get a concrete positive example.

It deserves a serious referee because the claim is precise, the construction is reproducible, and the algebraic verification is self-contained.

Referee Report

0 major / 2 minor

Summary. The paper defines a two-sided correlation kernel C_f on a finite group Γ for real-valued f by C_f(x,y) = |Γ|^{-1} ∑_{xa1=a2y} f(a1)f(a2) = E_z f(x^{-1}z)f(zy^{-1}). It proves that for every finite directed graph F, t(F, C_f) ≥ t(→K₂, C_f)^{e(F)} = (E_g f(g))^{2e(F)}. Equivalently, the directed 1-subdivision of F satisfies the Sidorenko lower bound in the product Cayley kernel W_f^×(x,y)=f(xy).

Significance. If the result holds, it verifies a Sidorenko-type inequality for an explicit family of directed kernels constructed from group correlations, with the bound holding for arbitrary real f and reducing exactly to the square of the average of f. The algebraic proof exploits left/right multiplication bijectivity on finite groups and supplies a clean, parameter-free verification for all directed 1-subdivisions in the multiplicative Cayley setting. This adds a new algebraic source of examples to the literature on Sidorenko inequalities.

minor comments (2)

[Abstract] Abstract: the main claim is stated clearly but the location of the proof within the manuscript is not indicated; a single sentence directing the reader to the relevant section would improve navigation.
[Definition of C_f] Definition of C_f (displayed equation): the equality of the two expressions for C_f is asserted via change-of-variables, but a one-line remark on the substitution z = xa1 (or equivalent) would make the equivalence immediate for readers outside the immediate subfield.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, accurate summary of the main result, and recommendation for minor revision. No major comments appear in the report.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper defines C_f explicitly via the two equivalent expressions involving sums over the finite group and proves the homomorphism density inequality directly from this definition. The equality t(→K₂, C_f) = (E f(g))^{2} follows from a change-of-variables computation that exploits the bijectivity of left and right multiplication; this is an algebraic identity, not a fit or self-reference. The general inequality for arbitrary directed F is presented as a new algebraic consequence of the correlation kernel's positivity/contraction properties, with no load-bearing self-citations, ansatzes imported from prior work, or renaming of known results. The derivation chain is self-contained and does not reduce any claimed prediction to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The argument rests on the standard definition of homomorphism density and the algebraic properties of finite groups used to equate the two forms of C_f; no free parameters or new physical entities are introduced.

axioms (1)

standard math Finite groups are closed under multiplication and inversion, allowing the sum over a1, a2 with xa1 = a2 y to be well-defined.
Invoked in the definition of C_f in the abstract.

invented entities (1)

Directed kernel C_f no independent evidence
purpose: To provide a concrete kernel on groups that satisfies the stated homomorphism-density inequality.
Newly defined in the paper; no independent evidence outside the construction is supplied.

pith-pipeline@v0.9.1-grok · 5773 in / 1370 out tokens · 39130 ms · 2026-06-26T08:16:32.591629+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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Graphs and Combinatorics , volume=

A correlation inequality for bipartite graphs , author=. Graphs and Combinatorics , volume=. 1993 , publisher=

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Diskretnaya Matematika , volume=

Inequalities for functionals generated by bipartite graphs , author=. Diskretnaya Matematika , volume=. 1991 , publisher=

1991

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arXiv preprint arXiv:1504.00858 , year=

Sparse graph limits, entropy maximization and transitive graphs , author=. arXiv preprint arXiv:1504.00858 , year=

Pith/arXiv arXiv

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Hatami, Hamed , journal=. Graph norms and. 2010 , publisher=

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