{"title":"二维$二维$库仑气体涡旋波动的定量界和整数值高斯自由场最大值","authors":"Christophe Garban, Avelio Sepúlveda","doi":"10.1112/plms.12551","DOIUrl":null,"url":null,"abstract":"In this paper, we study the influence of the vortices on the fluctuations of <math altimg=\"urn:x-wiley:00246115:media:plms12551:plms12551-math-0003\" display=\"inline\" location=\"graphic/plms12551-math-0003.png\">\n<semantics>\n<mrow>\n<mn>2</mn>\n<mi>d</mi>\n</mrow>\n$2d$</annotation>\n</semantics></math> systems such as the Coulomb gas, the Villain model, or the integer-valued Gaussian free field (GFF). In the case of the <math altimg=\"urn:x-wiley:00246115:media:plms12551:plms12551-math-0004\" display=\"inline\" location=\"graphic/plms12551-math-0004.png\">\n<semantics>\n<mrow>\n<mn>2</mn>\n<mi>d</mi>\n</mrow>\n$2d$</annotation>\n</semantics></math> Villain model, we prove that the fluctuations induced by the vortices are at least of the same order of magnitude as the ones produced by the spin wave. We obtain the following quantitative upper bound on the two-point correlation in <math altimg=\"urn:x-wiley:00246115:media:plms12551:plms12551-math-0005\" display=\"inline\" location=\"graphic/plms12551-math-0005.png\">\n<semantics>\n<msup>\n<mi mathvariant=\"double-struck\">Z</mi>\n<mn>2</mn>\n</msup>\n$\\mathbb {Z}^2$</annotation>\n</semantics></math> when <math altimg=\"urn:x-wiley:00246115:media:plms12551:plms12551-math-0006\" display=\"inline\" location=\"graphic/plms12551-math-0006.png\">\n<semantics>\n<mrow>\n<mi>β</mi>\n<mo>></mo>\n<mn>1</mn>\n</mrow>\n$\\beta >1$</annotation>\n</semantics></math>","PeriodicalId":49667,"journal":{"name":"Proceedings of the London Mathematical Society","volume":"27 1","pages":""},"PeriodicalIF":1.5000,"publicationDate":"2023-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"10","resultStr":"{\"title\":\"Quantitative bounds on vortex fluctuations in 2d$2d$ Coulomb gas and maximum of the integer-valued Gaussian free field\",\"authors\":\"Christophe Garban, Avelio Sepúlveda\",\"doi\":\"10.1112/plms.12551\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"In this paper, we study the influence of the vortices on the fluctuations of <math altimg=\\\"urn:x-wiley:00246115:media:plms12551:plms12551-math-0003\\\" display=\\\"inline\\\" location=\\\"graphic/plms12551-math-0003.png\\\">\\n<semantics>\\n<mrow>\\n<mn>2</mn>\\n<mi>d</mi>\\n</mrow>\\n$2d$</annotation>\\n</semantics></math> systems such as the Coulomb gas, the Villain model, or the integer-valued Gaussian free field (GFF). In the case of the <math altimg=\\\"urn:x-wiley:00246115:media:plms12551:plms12551-math-0004\\\" display=\\\"inline\\\" location=\\\"graphic/plms12551-math-0004.png\\\">\\n<semantics>\\n<mrow>\\n<mn>2</mn>\\n<mi>d</mi>\\n</mrow>\\n$2d$</annotation>\\n</semantics></math> Villain model, we prove that the fluctuations induced by the vortices are at least of the same order of magnitude as the ones produced by the spin wave. We obtain the following quantitative upper bound on the two-point correlation in <math altimg=\\\"urn:x-wiley:00246115:media:plms12551:plms12551-math-0005\\\" display=\\\"inline\\\" location=\\\"graphic/plms12551-math-0005.png\\\">\\n<semantics>\\n<msup>\\n<mi mathvariant=\\\"double-struck\\\">Z</mi>\\n<mn>2</mn>\\n</msup>\\n$\\\\mathbb {Z}^2$</annotation>\\n</semantics></math> when <math altimg=\\\"urn:x-wiley:00246115:media:plms12551:plms12551-math-0006\\\" display=\\\"inline\\\" location=\\\"graphic/plms12551-math-0006.png\\\">\\n<semantics>\\n<mrow>\\n<mi>β</mi>\\n<mo>></mo>\\n<mn>1</mn>\\n</mrow>\\n$\\\\beta >1$</annotation>\\n</semantics></math>\",\"PeriodicalId\":49667,\"journal\":{\"name\":\"Proceedings of the London Mathematical Society\",\"volume\":\"27 1\",\"pages\":\"\"},\"PeriodicalIF\":1.5000,\"publicationDate\":\"2023-07-23\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"10\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Proceedings of the London Mathematical Society\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1112/plms.12551\",\"RegionNum\":1,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Proceedings of the London Mathematical Society","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1112/plms.12551","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
Quantitative bounds on vortex fluctuations in 2d$2d$ Coulomb gas and maximum of the integer-valued Gaussian free field
In this paper, we study the influence of the vortices on the fluctuations of systems such as the Coulomb gas, the Villain model, or the integer-valued Gaussian free field (GFF). In the case of the Villain model, we prove that the fluctuations induced by the vortices are at least of the same order of magnitude as the ones produced by the spin wave. We obtain the following quantitative upper bound on the two-point correlation in when
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