Potential Theory and the Boundary of Combinatorial Graphs
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866915414528753664 |
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| author | Steinerberger, Stefan |
| author_facet | Steinerberger, Stefan |
| contents | Let $G=(V,E)$ be a finite, connected graph. We investigate a notion of boundary $\partial G \subseteq V$ and argue that it is well behaved from the point of view of potential theory. This is done by proving a number of discrete analogous of classical results for compact domains $Ω\subset \mathbb{R}^d$. These include (1) an analogue of Pólya's result that a random walk in $Ω$ typically hits the boundary $\partial Ω$ within $\lesssim \mbox{diam}(Ω)^2$ units of time, (2) an analogue of the Faber-Krahn inequality, (3) an analogue of the Hardy inequality, (4) an analogue of the Alexandrov-Bakelman-Pucci estimate, (5) a stability estimate for hot spots and (6) a Theorem of Björck stating that probability measures $μ$ that maximize $\int_{Ω\times Ω} \|x-y\|^α dμ(x) dμ(y)$ are fully supported in the boundary. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_20833 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Potential Theory and the Boundary of Combinatorial Graphs Steinerberger, Stefan Classical Analysis and ODEs Analysis of PDEs Combinatorics Probability Let $G=(V,E)$ be a finite, connected graph. We investigate a notion of boundary $\partial G \subseteq V$ and argue that it is well behaved from the point of view of potential theory. This is done by proving a number of discrete analogous of classical results for compact domains $Ω\subset \mathbb{R}^d$. These include (1) an analogue of Pólya's result that a random walk in $Ω$ typically hits the boundary $\partial Ω$ within $\lesssim \mbox{diam}(Ω)^2$ units of time, (2) an analogue of the Faber-Krahn inequality, (3) an analogue of the Hardy inequality, (4) an analogue of the Alexandrov-Bakelman-Pucci estimate, (5) a stability estimate for hot spots and (6) a Theorem of Björck stating that probability measures $μ$ that maximize $\int_{Ω\times Ω} \|x-y\|^α dμ(x) dμ(y)$ are fully supported in the boundary. |
| title | Potential Theory and the Boundary of Combinatorial Graphs |
| topic | Classical Analysis and ODEs Analysis of PDEs Combinatorics Probability |
| url | https://arxiv.org/abs/2507.20833 |