The optimal hypercontractive constants for ${\mathbb{Z}}_3$ and biased Bernoulli random variables

Abstract

We resolve a folklore problem of determining the optimal hypercontractive constants $r_{p,q}({\mathbb{Z}}_3)$ for the cyclic group ${\mathbb{Z}}_3$ for all $1<p<q<\infty$. More precisely, we have \[ r_{p,q}(\mathbb{Z}_3) = \frac{(1 + 2x)(1 - y)}{(1 + 2y)(1 - x)}, \] where $(x,y)$ is the unique solution in the open unit square $(0,1)\times (0,1)$ to the system of equations \begin{align*} \left\{ \begin{aligned} &\frac{1}{1+2x}\Big(\frac{1+2x^p}{3}\Big)^{\frac{1}{p}}=\frac{1}{1+2y}\Big(\frac{1+2y^q}{3}\Big)^{\frac{1}{q}},\\ &\frac{(1-x)(1-x^{p-1})}{1+2x^p}=\frac{(1-y)(1-y^{q-1})}{1+2y^q}. \end{aligned} \right. \end{align*} Consequently, for rational $p,q\in \mathbb{Q}$, the constants $r_{p,q}({\mathbb{Z}}_3)$ are algebraic numbers which generally admit no radical expressions, since their often rather complicated minimal polynomials may have non-solvable Galois groups. Our formalism relies on a key observation: the existence of nontrivial critical extremizers. This approach can also be adapted to resolve a long-standing open problem -- determining all optimal $(p,q)$-hypercontractive constants for biased Bernoulli random variables, which are closely related to noise operators. Several noteworthy phenomena emerge from numerical simulations: the monotonicity of the hypercontractive constants in the parameters, and the appearance of intriguing limit shapes. These phenomena merit further investigation.

We added the optimal hypercontractivity of biased Bernoulli random variables. 52 pages with 16 figures, 2 tables