A relation between isoperimetry and total variation decay with applications to graphs of non-negative Ollivier-Ricci curvature
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866912109572390912 |
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| author | Hutchcroft, Tom Lopez, Isaac M. |
| author_facet | Hutchcroft, Tom Lopez, Isaac M. |
| contents | We prove an inequality relating the isoperimetric profile of a graph to the decay of the random walk total variation distance $\sup_{x\sim y} ||P^n(x,\cdot)-P^n(y,\cdot)||_{\mathrm{TV}}$. This inequality implies a quantitative version of a theorem of Salez (GAFA 2022) stating that bounded-degree graphs of non-negative Ollivier-Ricci curvature cannot be expanders. Along the way, we prove universal upper-tail estimates for the random walk displacement $d(X_0,X_n)$ and information $-\log P^n(X_0,X_n)$, which may be of independent interest. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_04988 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A relation between isoperimetry and total variation decay with applications to graphs of non-negative Ollivier-Ricci curvature Hutchcroft, Tom Lopez, Isaac M. Probability Differential Geometry We prove an inequality relating the isoperimetric profile of a graph to the decay of the random walk total variation distance $\sup_{x\sim y} ||P^n(x,\cdot)-P^n(y,\cdot)||_{\mathrm{TV}}$. This inequality implies a quantitative version of a theorem of Salez (GAFA 2022) stating that bounded-degree graphs of non-negative Ollivier-Ricci curvature cannot be expanders. Along the way, we prove universal upper-tail estimates for the random walk displacement $d(X_0,X_n)$ and information $-\log P^n(X_0,X_n)$, which may be of independent interest. |
| title | A relation between isoperimetry and total variation decay with applications to graphs of non-negative Ollivier-Ricci curvature |
| topic | Probability Differential Geometry |
| url | https://arxiv.org/abs/2411.04988 |