On the number of excursion sets of planar Gaussian fields
The Nazarov–Sodin constant describes the average number of nodal set components of smooth Gaussian fields on large scales. We generalise this to a functional describing the corresponding number of level set components for arbitrary levels. Using results from Morse theory, we express this functional...
| Main Authors: | , , |
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| Format: | Journal article |
| Language: | English |
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Springer
2020
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| _version_ | 1797068910627913728 |
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| author | Beliaev, D McAuley, M Muirhead, S |
| author_facet | Beliaev, D McAuley, M Muirhead, S |
| author_sort | Beliaev, D |
| collection | OXFORD |
| description | The Nazarov–Sodin constant describes the average number of nodal set components of smooth Gaussian fields on large scales. We generalise this to a functional describing the corresponding number of level set components for arbitrary levels. Using results from Morse theory, we express this functional as an integral over the level densities of different types of critical points, and as a result deduce the absolute continuity of the functional as the level varies. We further give upper and lower bounds showing that the functional is at least bimodal for certain isotropic fields, including the important special case of the random plane wave. |
| first_indexed | 2024-03-06T22:16:48Z |
| format | Journal article |
| id | oxford-uuid:53a56bab-ed05-492b-97a6-e6d1f7f1b9ac |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-06T22:16:48Z |
| publishDate | 2020 |
| publisher | Springer |
| record_format | dspace |
| spelling | oxford-uuid:53a56bab-ed05-492b-97a6-e6d1f7f1b9ac2022-03-26T16:33:00ZOn the number of excursion sets of planar Gaussian fieldsJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:53a56bab-ed05-492b-97a6-e6d1f7f1b9acEnglishSymplectic ElementsSpringer2020Beliaev, DMcAuley, MMuirhead, SThe Nazarov–Sodin constant describes the average number of nodal set components of smooth Gaussian fields on large scales. We generalise this to a functional describing the corresponding number of level set components for arbitrary levels. Using results from Morse theory, we express this functional as an integral over the level densities of different types of critical points, and as a result deduce the absolute continuity of the functional as the level varies. We further give upper and lower bounds showing that the functional is at least bimodal for certain isotropic fields, including the important special case of the random plane wave. |
| spellingShingle | Beliaev, D McAuley, M Muirhead, S On the number of excursion sets of planar Gaussian fields |
| title | On the number of excursion sets of planar Gaussian fields |
| title_full | On the number of excursion sets of planar Gaussian fields |
| title_fullStr | On the number of excursion sets of planar Gaussian fields |
| title_full_unstemmed | On the number of excursion sets of planar Gaussian fields |
| title_short | On the number of excursion sets of planar Gaussian fields |
| title_sort | on the number of excursion sets of planar gaussian fields |
| work_keys_str_mv | AT beliaevd onthenumberofexcursionsetsofplanargaussianfields AT mcauleym onthenumberofexcursionsetsofplanargaussianfields AT muirheads onthenumberofexcursionsetsofplanargaussianfields |