Sharp uniform stability of the logarithmic Sobolev inequality on finite cycles and stability of the cubic Sobolev inequality
Abstract
We determine the sharp uniform constant in the quartic stability estimate for the logarithmic Sobolev inequality on finite cycles. We equip Cn=ℤ/nℤ with the uniform probability measure, and we let ℰn be the Dirichlet form of the simple random walk on Cn, normalized so that its spectral gap is one. Frank and Ivanisvili recently proved the sharp logarithmic Sobolev inequality 2ℰn(u)≥ Ent(u2) for all n≥ 4. We prove that 2ℰn(u)−Ent(u2)≥ 1/3‖ u−1‖ 24 for all n≥ 4 and all nonnegative u with ‖ u‖ 2=1. This is the sharp form, uniformly in n, of the quartic stability estimate of Xie and Zhang, who obtained the constant 1/12. The same estimate holds on the circle and for the word-length energy on Cn. Among these models, the four-cycle is the only extremal one: on C4 the constant 1/3 is sharp but not attained, while on Cn with n≥ 5 and on the circle the sharp constant is at least 1/3+c, where c>0 does not depend on n. We also give explicit upper bounds for these sharp constants. Our second main result is a stability estimate for the cubic Sobolev inequality of Frank and Ivanisvili, which is the main step in their proof. It gives the quantitative version of this inequality expected by Frank and Ivanisvili, with the optimal powers of ‖ u−1‖ 2: four on Cn with n≥ 5 and on the circle, and six on C4. Two key new ingredients of the proofs are a bound for the first Fourier coefficient of u in terms of its mean, which holds because u is nonnegative, and a sharp stability estimate for Gross's two-point logarithmic Sobolev inequality, with sharp constants in both the fourth and the sixth powers of the distance.
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BibTeX
@misc{chen2026sharp,
title = {{Sharp uniform stability of the logarithmic Sobolev inequality on finite cycles and stability of the cubic Sobolev inequality}},
author = {Lu Chen and Nguyen Lam and Guozhen Lu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.05347},
eprint = {2610.05347},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.05347},
}
RIS
TY - UNPB
TI - Sharp uniform stability of the logarithmic Sobolev inequality on finite cycles and stability of the cubic Sobolev inequality
AU - Lu Chen
AU - Nguyen Lam
AU - Guozhen Lu
PY - 2026
DA - 2026/10/04
PB - arXiv
DO - 10.48550/arxiv.2610.05347
UR - https://arxiv.org/abs/2610.05347
AB - We determine the sharp uniform constant in the quartic stability estimate for the logarithmic Sobolev inequality on finite cycles. We equip C_n=ℤ/nℤ with the uniform probability measure, and we let ℰ_n be the Dirichlet form of the simple random walk on C_n, normalized so that its spectral gap is one. Frank and Ivanisvili recently proved the sharp logarithmic Sobolev inequality 2ℰ_n(u)≥ Ent(u^2) for all n≥ 4. We prove that 2ℰ_n(u)−Ent(u^2)≥ 1/3‖ u−1‖ _2^4 for all n≥ 4 and all nonnegative u with ‖ u‖ _2=1. This is the sharp form, uniformly in n, of the quartic stability estimate of Xie and Zhang, who obtained the constant 1/12. The same estimate holds on the circle and for the word-length energy on C_n. Among these models, the four-cycle is the only extremal one: on C_4 the constant 1/3 is sharp but not attained, while on C_n with n≥ 5 and on the circle the sharp constant is at least 1/3+c, where c>0 does not depend on n. We also give explicit upper bounds for these sharp constants. Our second main result is a stability estimate for the cubic Sobolev inequality of Frank and Ivanisvili, which is the main step in their proof. It gives the quantitative version of this inequality expected by Frank and Ivanisvili, with the optimal powers of ‖ u−1‖ _2: four on C_n with n≥ 5 and on the circle, and six on C_4. Two key new ingredients of the proofs are a bound for the first Fourier coefficient of u in terms of its mean, which holds because u is nonnegative, and a sharp stability estimate for Gross's two-point logarithmic Sobolev inequality, with sharp constants in both the fourth and the sixth powers of the distance.
ER -