Werner's Measure on Self-Avoiding Loops and Welding
Werner's conformally invariant family of measures on self-avoiding loops on Riemann surfaces is determined by a single measure μ0 on self-avoiding loops in C∖{0} which surround 0. Our first major objective is to show that the measure μ0 is infinitesimally invariant with respect to conformal vec...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2014 |
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| Format: | Artikel |
| Sprache: | Englisch |
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Інститут математики НАН України
2014
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| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/146622 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Werner's Measure on Self-Avoiding Loops and Welding / A. Chavez, D. Pickrell // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 17 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862575750484328448 |
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| author | Chavez, A. Pickrell, D. |
| author_facet | Chavez, A. Pickrell, D. |
| citation_txt | Werner's Measure on Self-Avoiding Loops and Welding / A. Chavez, D. Pickrell // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 17 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | Werner's conformally invariant family of measures on self-avoiding loops on Riemann surfaces is determined by a single measure μ0 on self-avoiding loops in C∖{0} which surround 0. Our first major objective is to show that the measure μ0 is infinitesimally invariant with respect to conformal vector fields (essentially the Virasoro algebra of conformal field theory). This makes essential use of classical variational formulas of Duren and Schiffer, which we recast in representation theoretic terms for efficient computation. We secondly show how these formulas can be used to calculate (in principle, and sometimes explicitly) quantities (such as moments for coefficients of univalent functions) associated to the conformal welding for a self-avoiding loop. This gives an alternate proof of the uniqueness of Werner's measure. We also attempt to use these variational formulas to derive a differential equation for the (Laplace transform of) the ''diagonal distribution'' for the conformal welding associated to a loop; this generalizes in a suggestive way to a deformation of Werner's measure conjectured to exist by Kontsevich and Suhov (a basic inspiration for this paper).
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| first_indexed | 2025-11-26T13:21:51Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-146622 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-11-26T13:21:51Z |
| publishDate | 2014 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Chavez, A. Pickrell, D. 2019-02-10T10:12:47Z 2019-02-10T10:12:47Z 2014 Werner's Measure on Self-Avoiding Loops and Welding / A. Chavez, D. Pickrell // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 17 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 60D05; 60B15; 17B68; 30C99 DOI:10.3842/SIGMA.2014.081 https://nasplib.isofts.kiev.ua/handle/123456789/146622 Werner's conformally invariant family of measures on self-avoiding loops on Riemann surfaces is determined by a single measure μ0 on self-avoiding loops in C∖{0} which surround 0. Our first major objective is to show that the measure μ0 is infinitesimally invariant with respect to conformal vector fields (essentially the Virasoro algebra of conformal field theory). This makes essential use of classical variational formulas of Duren and Schiffer, which we recast in representation theoretic terms for efficient computation. We secondly show how these formulas can be used to calculate (in principle, and sometimes explicitly) quantities (such as moments for coefficients of univalent functions) associated to the conformal welding for a self-avoiding loop. This gives an alternate proof of the uniqueness of Werner's measure. We also attempt to use these variational formulas to derive a differential equation for the (Laplace transform of) the ''diagonal distribution'' for the conformal welding associated to a loop; this generalizes in a suggestive way to a deformation of Werner's measure conjectured to exist by Kontsevich and Suhov (a basic inspiration for this paper). We thank Tom Kennedy for useful conversations, and the referees for many useful suggestions regarding exposition and inclusion of references. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Werner's Measure on Self-Avoiding Loops and Welding Article published earlier |
| spellingShingle | Werner's Measure on Self-Avoiding Loops and Welding Chavez, A. Pickrell, D. |
| title | Werner's Measure on Self-Avoiding Loops and Welding |
| title_full | Werner's Measure on Self-Avoiding Loops and Welding |
| title_fullStr | Werner's Measure on Self-Avoiding Loops and Welding |
| title_full_unstemmed | Werner's Measure on Self-Avoiding Loops and Welding |
| title_short | Werner's Measure on Self-Avoiding Loops and Welding |
| title_sort | werner's measure on self-avoiding loops and welding |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/146622 |
| work_keys_str_mv | AT chaveza wernersmeasureonselfavoidingloopsandwelding AT pickrelld wernersmeasureonselfavoidingloopsandwelding |