A Test of a Conjecture of Cardy
In reference to Werner's measure on self-avoiding loops on Riemann surfaces, Cardy conjectured a formula for the measure of all homotopically nontrivial loops in a finite type annular region with modular parameter . Ang, Remy, and Sun have announced a proof of this conjecture using random confo...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2025 |
| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | Englisch |
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Інститут математики НАН України
2025
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| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/213181 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | A Test of a Conjecture of Cardy. Van Higgs and Doug Pickrell. SIGMA 21 (2025), 034, 12 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862575423893798912 |
|---|---|
| author | Higgs, Van Pickrell, Doug |
| author_facet | Higgs, Van Pickrell, Doug |
| citation_txt | A Test of a Conjecture of Cardy. Van Higgs and Doug Pickrell. SIGMA 21 (2025), 034, 12 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | In reference to Werner's measure on self-avoiding loops on Riemann surfaces, Cardy conjectured a formula for the measure of all homotopically nontrivial loops in a finite type annular region with modular parameter . Ang, Remy, and Sun have announced a proof of this conjecture using random conformal geometry. Cardy's formula implies that the measure of the set of homotopically nontrivial loops in the punctured plane that intersect ¹ equals 2π/√3. This set is the disjoint union of the set of loops that avoid a ray from the unit circle to infinity and its complement. There is an inclusion/exclusion sum which, in the limit, calculates the measure of the set of loops that avoid a ray. Each term in the sum involves finding the transfinite diameter of a slit domain. This is numerically accessible using the remarkable Schwarz-Christoffel package developed by Driscoll and Trefethen. Our calculations suggest this sum is around π, consistent with Cardy's formula.
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| first_indexed | 2026-03-21T11:59:14Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-213181 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-21T11:59:14Z |
| publishDate | 2025 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Higgs, Van Pickrell, Doug 2026-02-16T16:34:07Z 2025 A Test of a Conjecture of Cardy. Van Higgs and Doug Pickrell. SIGMA 21 (2025), 034, 12 pages 1815-0659 2020 Mathematics Subject Classification: 60J67; 30C20; 65E10 arXiv:2401.03600 https://nasplib.isofts.kiev.ua/handle/123456789/213181 https://doi.org/10.3842/SIGMA.2025.034 In reference to Werner's measure on self-avoiding loops on Riemann surfaces, Cardy conjectured a formula for the measure of all homotopically nontrivial loops in a finite type annular region with modular parameter . Ang, Remy, and Sun have announced a proof of this conjecture using random conformal geometry. Cardy's formula implies that the measure of the set of homotopically nontrivial loops in the punctured plane that intersect ¹ equals 2π/√3. This set is the disjoint union of the set of loops that avoid a ray from the unit circle to infinity and its complement. There is an inclusion/exclusion sum which, in the limit, calculates the measure of the set of loops that avoid a ray. Each term in the sum involves finding the transfinite diameter of a slit domain. This is numerically accessible using the remarkable Schwarz-Christoffel package developed by Driscoll and Trefethen. Our calculations suggest this sum is around π, consistent with Cardy's formula. We thank Toby Driscoll for help with using the SC package for slit domains, and the referees for comments that improved the exposition. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications A Test of a Conjecture of Cardy Article published earlier |
| spellingShingle | A Test of a Conjecture of Cardy Higgs, Van Pickrell, Doug |
| title | A Test of a Conjecture of Cardy |
| title_full | A Test of a Conjecture of Cardy |
| title_fullStr | A Test of a Conjecture of Cardy |
| title_full_unstemmed | A Test of a Conjecture of Cardy |
| title_short | A Test of a Conjecture of Cardy |
| title_sort | test of a conjecture of cardy |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/213181 |
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