Modified logarithmic Sobolev inequality for 1D non-commuting Hamiltonians
Abstract
Given a finite-range, non-commuting Hamiltonian on a 1D chain of finite spins, we prove a modified logarithmic Sobolev inequality with constant Ω(1/log n) for the heat-bath dynamics, a quasi-local Gibbs sampler, and for the regularised heat-bath dynamics. This implies rapid mixing of both samplers towards the Gibbs state in relative entropy and trace norm, with mixing times of order 𝒪(log n· log(n/ε)). The interactions may be spatially inhomogeneous. The proof combines weak quasi-factorisation of relative entropy, decay-of-correlation estimates, a uniform conditional local gap, and a buffered comparison between conditional entropy and quadratic energy. The resulting bounded entropy defect is removed using the global gap. To the best of our knowledge, this is the first proof of MLSI in the non-commuting regime.
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BibTeX
@misc{capel2026modified,
title = {{Modified logarithmic Sobolev inequality for 1D non-commuting Hamiltonians}},
author = {Ángela Capel and David Pérez-García and Matteo Scandi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.03683},
eprint = {2610.03683},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.03683},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Modified logarithmic Sobolev inequality for 1D non-commuting Hamiltonians
AU - Ángela Capel
AU - David Pérez-García
AU - Matteo Scandi
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.03683
UR - https://arxiv.org/abs/2610.03683
AB - Given a finite-range, non-commuting Hamiltonian on a 1D chain of finite spins, we prove a modified logarithmic Sobolev inequality with constant Ω(1/log n) for the heat-bath dynamics, a quasi-local Gibbs sampler, and for the regularised heat-bath dynamics. This implies rapid mixing of both samplers towards the Gibbs state in relative entropy and trace norm, with mixing times of order 𝒪(log n· log(n/ε)). The interactions may be spatially inhomogeneous. The proof combines weak quasi-factorisation of relative entropy, decay-of-correlation estimates, a uniform conditional local gap, and a buffered comparison between conditional entropy and quadratic energy. The resulting bounded entropy defect is removed using the global gap. To the best of our knowledge, this is the first proof of MLSI in the non-commuting regime.
ER -