UNIFORM SPANNING FORESTS OF PLANAR GRAPHS
We prove that the free uniform spanning forest of any bounded degree proper plane graph is connected almost surely, answering a question of Benjamini, Lyons, Peres and Schramm. We provide a quantitative form of this result, calculating the critical exponents governing the geometry of the uniform spa...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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Cambridge University Press
2019-01-01
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| Series: | Forum of Mathematics, Sigma |
| Subjects: | |
| Online Access: | https://www.cambridge.org/core/product/identifier/S2050509419000148/type/journal_article |
| _version_ | 1827994839637032960 |
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| author | TOM HUTCHCROFT ASAF NACHMIAS |
| author_facet | TOM HUTCHCROFT ASAF NACHMIAS |
| author_sort | TOM HUTCHCROFT |
| collection | DOAJ |
| description | We prove that the free uniform spanning forest of any bounded degree proper plane graph is connected almost surely, answering a question of Benjamini, Lyons, Peres and Schramm. We provide a quantitative form of this result, calculating the critical exponents governing the geometry of the uniform spanning forests of transient proper plane graphs with bounded degrees and codegrees. We find that the same exponents hold universally over this entire class of graphs provided that measurements are made using the hyperbolic geometry of their circle packings rather than their usual combinatorial geometry. |
| first_indexed | 2024-04-10T04:47:12Z |
| format | Article |
| id | doaj.art-6eaedc0c98d545238e6da3603712caf8 |
| institution | Directory Open Access Journal |
| issn | 2050-5094 |
| language | English |
| last_indexed | 2024-04-10T04:47:12Z |
| publishDate | 2019-01-01 |
| publisher | Cambridge University Press |
| record_format | Article |
| series | Forum of Mathematics, Sigma |
| spelling | doaj.art-6eaedc0c98d545238e6da3603712caf82023-03-09T12:34:44ZengCambridge University PressForum of Mathematics, Sigma2050-50942019-01-01710.1017/fms.2019.14UNIFORM SPANNING FORESTS OF PLANAR GRAPHSTOM HUTCHCROFT0ASAF NACHMIAS1Statslab, DPMMS, University of Cambridge, Cambridge CB3 0WB, UK;Department of Mathematical Sciences, Tel Aviv University, Tel Aviv 6997801, Israel;We prove that the free uniform spanning forest of any bounded degree proper plane graph is connected almost surely, answering a question of Benjamini, Lyons, Peres and Schramm. We provide a quantitative form of this result, calculating the critical exponents governing the geometry of the uniform spanning forests of transient proper plane graphs with bounded degrees and codegrees. We find that the same exponents hold universally over this entire class of graphs provided that measurements are made using the hyperbolic geometry of their circle packings rather than their usual combinatorial geometry.https://www.cambridge.org/core/product/identifier/S2050509419000148/type/journal_article60D05 (primary)05C10 (secondary) |
| spellingShingle | TOM HUTCHCROFT ASAF NACHMIAS UNIFORM SPANNING FORESTS OF PLANAR GRAPHS Forum of Mathematics, Sigma 60D05 (primary) 05C10 (secondary) |
| title | UNIFORM SPANNING FORESTS OF PLANAR GRAPHS |
| title_full | UNIFORM SPANNING FORESTS OF PLANAR GRAPHS |
| title_fullStr | UNIFORM SPANNING FORESTS OF PLANAR GRAPHS |
| title_full_unstemmed | UNIFORM SPANNING FORESTS OF PLANAR GRAPHS |
| title_short | UNIFORM SPANNING FORESTS OF PLANAR GRAPHS |
| title_sort | uniform spanning forests of planar graphs |
| topic | 60D05 (primary) 05C10 (secondary) |
| url | https://www.cambridge.org/core/product/identifier/S2050509419000148/type/journal_article |
| work_keys_str_mv | AT tomhutchcroft uniformspanningforestsofplanargraphs AT asafnachmias uniformspanningforestsofplanargraphs |