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arXiv · 2609.15553

Phase transition in optimal hypercontractivity

Jie Cao·Shilei Fan·Yong Han·Yanqi Qiu·Zipeng Wang
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Abstract

We discover an exponent-dependent phase transition phenomenon for optimal hypercontractivity: for every prescribed $q_0>2$, there exists a reversible continuous-time Markov chain on three state space with normalized spectral gap whose $(2,q)$-optimal hypercontractivity time satisfies $$ \text{$t_{\mathrm{opt}}(2,q)=\frac12\log(q-1)$ if and only if $q\ge q_0$},$$ whereas the strict inequality $t_{\mathrm{opt}}(2,q)>\frac12\log(q-1)$ holds for $2<q<q_0$.

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Jie Cao, Shilei Fan, Yong Han, Yanqi Qiu, Zipeng Wang. 2026-09-14. Phase transition in optimal hypercontractivity. https://arxiv.org/abs/2609.15553

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