{"title":"关于 Pólya 的随机行走常数","authors":"Robert Gaunt, Saralees Nadarajah, Tibor Pogány","doi":"10.1090/proc/16854","DOIUrl":null,"url":null,"abstract":"<p>A celebrated result in probability theory is that a simple symmetric random walk on the <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"d\"> <mml:semantics> <mml:mi>d</mml:mi> <mml:annotation encoding=\"application/x-tex\">d</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-dimensional lattice <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper Z Superscript d\"> <mml:semantics> <mml:msup> <mml:mrow> <mml:mi mathvariant=\"double-struck\">Z</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> <mml:annotation encoding=\"application/x-tex\">\\mathbb {Z}^d</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is recurrent for <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"d equals 1 comma 2\"> <mml:semantics> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">d=1,2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and transient for <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"d greater-than-or-equal-to 3\"> <mml:semantics> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">d\\geq 3</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. In this note, we derive a closed-form expression, in terms of the Lauricella function <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper F Subscript upper C\"> <mml:semantics> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>C</mml:mi> </mml:msub> <mml:annotation encoding=\"application/x-tex\">F_C</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, for the return probability for all <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"d greater-than-or-equal-to 3\"> <mml:semantics> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">d\\geq 3</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Previously, a closed-form formula had only been available for <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"d equals 3\"> <mml:semantics> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>=</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">d=3</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.</p>","PeriodicalId":20696,"journal":{"name":"Proceedings of the American Mathematical Society","volume":"22 1","pages":""},"PeriodicalIF":0.8000,"publicationDate":"2024-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On Pólya’s random walk constants\",\"authors\":\"Robert Gaunt, Saralees Nadarajah, Tibor Pogány\",\"doi\":\"10.1090/proc/16854\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>A celebrated result in probability theory is that a simple symmetric random walk on the <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"d\\\"> <mml:semantics> <mml:mi>d</mml:mi> <mml:annotation encoding=\\\"application/x-tex\\\">d</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-dimensional lattice <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"double-struck upper Z Superscript d\\\"> <mml:semantics> <mml:msup> <mml:mrow> <mml:mi mathvariant=\\\"double-struck\\\">Z</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> <mml:annotation encoding=\\\"application/x-tex\\\">\\\\mathbb {Z}^d</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is recurrent for <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"d equals 1 comma 2\\\"> <mml:semantics> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">d=1,2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and transient for <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"d greater-than-or-equal-to 3\\\"> <mml:semantics> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">d\\\\geq 3</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. In this note, we derive a closed-form expression, in terms of the Lauricella function <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper F Subscript upper C\\\"> <mml:semantics> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>C</mml:mi> </mml:msub> <mml:annotation encoding=\\\"application/x-tex\\\">F_C</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, for the return probability for all <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"d greater-than-or-equal-to 3\\\"> <mml:semantics> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">d\\\\geq 3</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Previously, a closed-form formula had only been available for <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"d equals 3\\\"> <mml:semantics> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>=</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">d=3</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.</p>\",\"PeriodicalId\":20696,\"journal\":{\"name\":\"Proceedings of the American Mathematical Society\",\"volume\":\"22 1\",\"pages\":\"\"},\"PeriodicalIF\":0.8000,\"publicationDate\":\"2024-04-19\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Proceedings of the American Mathematical Society\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1090/proc/16854\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Proceedings of the American Mathematical Society","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1090/proc/16854","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
概率论中一个著名的结果是，在 d d 维网格 Z d \mathbb {Z}^d 上的简单对称随机行走在 d = 1 , 2 d=1,2 时是经常性的，而在 d ≥ 3 d\geq 3 时是瞬时性的。在本说明中，我们用劳里切拉函数 F C F_C 为所有 d ≥ 3 d\geq 3 的回归概率推导出一个闭式表达式。在此之前，只有 d=3 d=3 时才有闭式公式。
A celebrated result in probability theory is that a simple symmetric random walk on the dd-dimensional lattice Zd\mathbb {Z}^d is recurrent for d=1,2d=1,2 and transient for d≥3d\geq 3. In this note, we derive a closed-form expression, in terms of the Lauricella function FCF_C, for the return probability for all d≥3d\geq 3. Previously, a closed-form formula had only been available for d=3d=3.
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