Gaussian concentration, integral probability metrics, and coupling functionals for infinite lattice systems
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866914406832537600 |
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| author | Chazottes, J. -R. Collet, P. Redig, F. |
| author_facet | Chazottes, J. -R. Collet, P. Redig, F. |
| contents | We develop a transport-entropy framework for Gaussian concentration inequalities on the infinite product space $S^{\mathbb Z^d}$, where $S$ is a finite set, in which sensitivity is measured by the $\ell^2$-norm of local oscillations.
We show that the associated transportation costs cannot be induced by any metric or cost function on the configuration space, due to a structural lack of extensivity in infinite product spaces.
Our main result proves that the associated integral probability metric and coupling functional coincide in finite volume, yielding a duality extending the classical Kantorovich-Rubinstein theorem beyond the metric setting. As a consequence, Marton's coupling inequality in all finite volumes is equivalent to Gaussian concentration, yielding a new characterization in the infinite-product setting.
In the translation-invariant setting, the corresponding metrics converge in the thermodynamic limit to the $\bar d$-metric. We further introduce a thermodynamic Gaussian concentration bound and prove its equivalence with a transport-entropy inequality involving the relative entropy density. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_17861 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Gaussian concentration, integral probability metrics, and coupling functionals for infinite lattice systems Chazottes, J. -R. Collet, P. Redig, F. Probability Mathematical Physics We develop a transport-entropy framework for Gaussian concentration inequalities on the infinite product space $S^{\mathbb Z^d}$, where $S$ is a finite set, in which sensitivity is measured by the $\ell^2$-norm of local oscillations. We show that the associated transportation costs cannot be induced by any metric or cost function on the configuration space, due to a structural lack of extensivity in infinite product spaces. Our main result proves that the associated integral probability metric and coupling functional coincide in finite volume, yielding a duality extending the classical Kantorovich-Rubinstein theorem beyond the metric setting. As a consequence, Marton's coupling inequality in all finite volumes is equivalent to Gaussian concentration, yielding a new characterization in the infinite-product setting. In the translation-invariant setting, the corresponding metrics converge in the thermodynamic limit to the $\bar d$-metric. We further introduce a thermodynamic Gaussian concentration bound and prove its equivalence with a transport-entropy inequality involving the relative entropy density. |
| title | Gaussian concentration, integral probability metrics, and coupling functionals for infinite lattice systems |
| topic | Probability Mathematical Physics |
| url | https://arxiv.org/abs/2603.17861 |