{"title":"局部一致随机置换极限曲面的连续性","authors":"Jonas Sjöstrand","doi":"10.1016/j.aam.2023.102636","DOIUrl":null,"url":null,"abstract":"<div><p>A locally uniform random permutation is generated by sampling <em>n</em> points independently from some absolutely continuous distribution <em>ρ</em> on the plane and interpreting them as a permutation by the rule that <em>i</em> maps to <em>j</em> if the <em>i</em>th point from the left is the <em>j</em>th point from below. As <em>n</em> tends to infinity, decreasing subsequences in the permutation will appear as curves in the plane, and by interpreting these as level curves, a union of decreasing subsequences gives rise to a surface. In a recent paper by the author it was shown that, for any <span><math><mi>r</mi><mo>≥</mo><mn>0</mn></math></span>, under the correct scaling as <em>n</em> tends to infinity, the surface of the largest union of <span><math><mo>⌊</mo><mi>r</mi><msqrt><mrow><mi>n</mi></mrow></msqrt><mo>⌋</mo></math></span> decreasing subsequences approaches a limit in the sense that it will come close to a maximizer of a specific variational integral (and, under reasonable assumptions, that the maximizer is essentially unique). In the present paper we show that there exists a continuous maximizer, provided that <em>ρ</em> has bounded density and support.</p><p>The key ingredient in the proof is a new theorem about real functions of two variables that are increasing in both variables: We show that, for any constant <em>C</em>, any such function can be made continuous without increasing the diameter of its image or decreasing anywhere the product of its partial derivatives clipped by <em>C</em>, that is the minimum of the product and <em>C</em>.</p></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.0000,"publicationDate":"2023-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0196885823001549/pdfft?md5=d27a29e2bb1f7d6c2151068a9fe577e3&pid=1-s2.0-S0196885823001549-main.pdf","citationCount":"0","resultStr":"{\"title\":\"Continuity of limit surfaces of locally uniform random permutations\",\"authors\":\"Jonas Sjöstrand\",\"doi\":\"10.1016/j.aam.2023.102636\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>A locally uniform random permutation is generated by sampling <em>n</em> points independently from some absolutely continuous distribution <em>ρ</em> on the plane and interpreting them as a permutation by the rule that <em>i</em> maps to <em>j</em> if the <em>i</em>th point from the left is the <em>j</em>th point from below. As <em>n</em> tends to infinity, decreasing subsequences in the permutation will appear as curves in the plane, and by interpreting these as level curves, a union of decreasing subsequences gives rise to a surface. In a recent paper by the author it was shown that, for any <span><math><mi>r</mi><mo>≥</mo><mn>0</mn></math></span>, under the correct scaling as <em>n</em> tends to infinity, the surface of the largest union of <span><math><mo>⌊</mo><mi>r</mi><msqrt><mrow><mi>n</mi></mrow></msqrt><mo>⌋</mo></math></span> decreasing subsequences approaches a limit in the sense that it will come close to a maximizer of a specific variational integral (and, under reasonable assumptions, that the maximizer is essentially unique). In the present paper we show that there exists a continuous maximizer, provided that <em>ρ</em> has bounded density and support.</p><p>The key ingredient in the proof is a new theorem about real functions of two variables that are increasing in both variables: We show that, for any constant <em>C</em>, any such function can be made continuous without increasing the diameter of its image or decreasing anywhere the product of its partial derivatives clipped by <em>C</em>, that is the minimum of the product and <em>C</em>.</p></div>\",\"PeriodicalId\":50877,\"journal\":{\"name\":\"Advances in Applied Mathematics\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.0000,\"publicationDate\":\"2023-11-28\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://www.sciencedirect.com/science/article/pii/S0196885823001549/pdfft?md5=d27a29e2bb1f7d6c2151068a9fe577e3&pid=1-s2.0-S0196885823001549-main.pdf\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Advances in Applied Mathematics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0196885823001549\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Advances in Applied Mathematics","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0196885823001549","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
Continuity of limit surfaces of locally uniform random permutations
A locally uniform random permutation is generated by sampling n points independently from some absolutely continuous distribution ρ on the plane and interpreting them as a permutation by the rule that i maps to j if the ith point from the left is the jth point from below. As n tends to infinity, decreasing subsequences in the permutation will appear as curves in the plane, and by interpreting these as level curves, a union of decreasing subsequences gives rise to a surface. In a recent paper by the author it was shown that, for any , under the correct scaling as n tends to infinity, the surface of the largest union of decreasing subsequences approaches a limit in the sense that it will come close to a maximizer of a specific variational integral (and, under reasonable assumptions, that the maximizer is essentially unique). In the present paper we show that there exists a continuous maximizer, provided that ρ has bounded density and support.
The key ingredient in the proof is a new theorem about real functions of two variables that are increasing in both variables: We show that, for any constant C, any such function can be made continuous without increasing the diameter of its image or decreasing anywhere the product of its partial derivatives clipped by C, that is the minimum of the product and C.
期刊介绍:
Interdisciplinary in its coverage, Advances in Applied Mathematics is dedicated to the publication of original and survey articles on rigorous methods and results in applied mathematics. The journal features articles on discrete mathematics, discrete probability theory, theoretical statistics, mathematical biology and bioinformatics, applied commutative algebra and algebraic geometry, convexity theory, experimental mathematics, theoretical computer science, and other areas.
Emphasizing papers that represent a substantial mathematical advance in their field, the journal is an excellent source of current information for mathematicians, computer scientists, applied mathematicians, physicists, statisticians, and biologists. Over the past ten years, Advances in Applied Mathematics has published research papers written by many of the foremost mathematicians of our time.