Hitting Times in the Binomial Random Graph

IF 1 3区数学 Q2 MATHEMATICS

Journal of Graph Theory Pub Date : 2026-08-11 Epub Date: 2026-05-22 DOI:10.1002/jgt.70057

Bertille Granet, Felix Joos, Jonathan Schrodt

{"title":"Hitting Times in the Binomial Random Graph","authors":"Bertille Granet, Felix Joos, Jonathan Schrodt","doi":"10.1002/jgt.70057","DOIUrl":null,"url":null,"abstract":"Fix <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>k</mi>\n \n <mo>≥</mo>\n \n <mn>2</mn>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt70057:jgt70057-math-0001\" wiley:location=\"equation/jgt70057-math-0001.png\"><mrow><mrow><mi>k</mi><mo>\\unicode{x02265}</mo><mn>2</mn></mrow></mrow></math></annotation>\n </semantics></math>, choose <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mfrac>\n <mrow>\n <mi>log</mi>\n \n <mi>n</mi>\n </mrow>\n \n <msup>\n <mi>n</mi>\n \n <mrow>\n <mrow>\n <mo>(</mo>\n \n <mrow>\n <mi>k</mi>\n \n <mo>−</mo>\n \n <mn>1</mn>\n </mrow>\n \n <mo>)</mo>\n </mrow>\n \n <mo>∕</mo>\n \n <mi>k</mi>\n </mrow>\n </msup>\n </mfrac>\n \n <mo>≤</mo>\n \n <mi>p</mi>\n \n <mo>≤</mo>\n \n <mn>1</mn>\n \n <mo>−</mo>\n \n <mi>Ω</mi>\n \n <mrow>\n <mo>(</mo>\n \n <mfrac>\n <mrow>\n <msup>\n <mi>log</mi>\n \n <mn>4</mn>\n </msup>\n \n <mi>n</mi>\n </mrow>\n \n <mi>n</mi>\n </mfrac>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt70057:jgt70057-math-0002\" wiley:location=\"equation/jgt70057-math-0002.png\"><mrow><mrow><mfrac><mrow><mi>log</mi><mi>n</mi></mrow><msup><mi>n</mi><mrow><mrow><mo>(</mo><mrow><mi>k</mi><mo>\\unicode{x02212}</mo><mn>1</mn></mrow><mo>)</mo></mrow><mo>\\unicode{x02215}</mo><mi>k</mi></mrow></msup></mfrac><mo>\\unicode{x02264}</mo><mi>p</mi><mo>\\unicode{x02264}</mo><mn>1</mn><mo>\\unicode{x02212}</mo><mi mathvariant=\"normal\">\\unicode{x003A9}</mi><mrow><mo stretchy=\"true\">(</mo><mfrac><mrow><msup><mi>log</mi><mn>4</mn></msup><mi>n</mi></mrow><mi>n</mi></mfrac><mo stretchy=\"true\">)</mo></mrow></mrow></mrow></math></annotation>\n </semantics></math>, and consider <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>G</mi>\n \n <mo>~</mo>\n \n <mi>G</mi>\n \n <mrow>\n <mo>(</mo>\n \n <mrow>\n <mi>n</mi>\n \n <mo>,</mo>\n \n <mi>p</mi>\n </mrow>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt70057:jgt70057-math-0003\" wiley:location=\"equation/jgt70057-math-0003.png\"><mrow><mrow><mi>G</mi><mo>\\unicode{x0007E}</mo><mi>G</mi><mrow><mo>(</mo><mrow><mi>n</mi><mo>,</mo><mi>p</mi></mrow><mo>)</mo></mrow></mrow></mrow></math></annotation>\n </semantics></math>. For any pair of vertices <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>v</mi>\n \n <mo>,</mo>\n \n <mi>w</mi>\n \n <mo>∈</mo>\n \n <mi>V</mi>\n \n <mrow>\n <mo>(</mo>\n \n <mi>G</mi>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt70057:jgt70057-math-0004\" wiley:location=\"equation/jgt70057-math-0004.png\"><mrow><mrow><mi>v</mi><mo>,</mo><mi>w</mi><mo>\\unicode{x02208}</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></mrow></math></annotation>\n </semantics></math>, we give a simple and precise formula for the expected number of steps that a random walk on <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>G</mi>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt70057:jgt70057-math-0005\" wiley:location=\"equation/jgt70057-math-0005.png\"><mrow><mrow><mi>G</mi></mrow></mrow></math></annotation>\n </semantics></math> starting at <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>w</mi>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt70057:jgt70057-math-0006\" wiley:location=\"equation/jgt70057-math-0006.png\"><mrow><mrow><mi>w</mi></mrow></mrow></math></annotation>\n </semantics></math> needs to first arrive at <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>v</mi>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt70057:jgt70057-math-0007\" wiley:location=\"equation/jgt70057-math-0007.png\"><mrow><mrow><mi>v</mi></mrow></mrow></math></annotation>\n </semantics></math>. The formula only depends on basic structural properties of <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>G</mi>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt70057:jgt70057-math-0008\" wiley:location=\"equation/jgt70057-math-0008.png\"><mrow><mrow><mi>G</mi></mrow></mrow></math></annotation>\n </semantics></math>. This improves and extends recent results of Ottolini and Steinerberger, as well as Ottolini, who considered this problem for constant as well as for mildly vanishing <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>p</mi>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt70057:jgt70057-math-0009\" wiley:location=\"equation/jgt70057-math-0009.png\"><mrow><mrow><mi>p</mi></mrow></mrow></math></annotation>\n </semantics></math>.","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"113 2","pages":"377-393"},"PeriodicalIF":1.0000,"publicationDate":"2026-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.70057","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Graph Theory","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/jgt.70057","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"2026/5/22 0:00:00","PubModel":"Epub","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}

引用次数: 0

Abstract

Fix $k \geq 2$ , choose $\frac{\log n}{n^{(k - 1) ∕ k}} \leq p \leq 1 - Ω (\frac{\log^{4} n}{n})$ , and consider $G ~ G (n, p)$ . For any pair of vertices $v, w \in V (G)$ , we give a simple and precise formula for the expected number of steps that a random walk on $G$ starting at $w$ needs to first arrive at $v$ . The formula only depends on basic structural properties of $G$ . This improves and extends recent results of Ottolini and Steinerberger, as well as Ottolini, who considered this problem for constant as well as for mildly vanishing $p$ .

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二项随机图中的命中次数

修正k≥2 &lt；math xmlns="http://www.w3.org/1998/Math/MathML" altimg=“urn:x-wiley:03649024:media:jgt70057:jgt70057-math-0001”魏:地方= "方程/ jgt70057-math-0001.png > < mrow&gt < mrow> &lt mi> k&lt / mi> < mo&gt \ unicode {x02265 < / mo> &lt mn> 2< / mn&gt < / mrow> &lt / mrow> &lt / math&gt ; ,选择日志n n (K−1)∕k≤p≤1−Ω (日志4 n n)数学xmlns="http://www.w3.org/1998/Math/MathML" altimg=“urn:x-wiley:03649024:media:jgt70057:jgt70057- math0002 ”魏:地方= "方程/ jgt70057-math-0002.png > < mrow&gt < mrow> &lt mfrac> < mrow&gt < mi> log&lt / mi> < mi&gt n< / mi> &lt / mrow> < msup&gt < mi> n&lt / mi> < mrow&gt < mrow> &lt mo> (&lt / mo> < mrow&gt < mi> k&lt / mi> < mo&gt \ unicode {x02212 < / mo> &lt mn> 1&lt / mn> &lt / mrow> < mo&gt) < / mo> &lt / mrow> < mo&gt \ unicode {x02215 < / mo&gt < mi> k&lt / mi> < / mrow&gt < / msup> &lt / mfrac> < mo&gt \ unicode {x。02264 &lt……;/ mo&gt < mi> p&lt / mi> < mo&gt \ unicode {x02264 < / mo> &lt mn> 1&lt / mn> < mo&gt \ unicode {x02212 < / mo&gt <弥mathvariant = > \ "正常" unicode {x003A9 < / mi> &lt mrow> &lt mo stretchy = >“真”(< / mo> &lt mfrac> < mrow&gt < msup> &lt mi> log< / mi&gt < mn> 4&lt / mn> < / msup&gt < mi> n&lt / mi> &lt / mrow> < mi&gt n< / mi> &lt / mfrac> &lt mo stretchy =“真”&lt >); / mo> &lt / mrow> < / mrow&gt < / mrow> &lt / math>，并考虑G ~ G (n，p) <math xmlns="http://www.w3.org/1998/Math/MathML" altimg=“urn:x-wiley:03649024:media:jgt70057:jgt70057- math0003 ”魏:地方= "方程/ jgt70057-math-0003.png > < mrow&gt < mrow> &lt mi> G&lt / mi> < mo&gt \ unicode {x0007E < / mo> &lt mi> G&lt / mi> < mrow&gt < mo> (&lt / mo> < mrow&gt < mi> n&lt / mi> < mo&gt < / mo> < mi&gt p< / mi> &lt / mrow> < mo&gt) < / mo> &lt / mrow> < / mrow&gt < / mrow> &lt / math&gt ; . 对于任意顶点v对，w∈V (G) <math xmlns=“http://www.w3.org/1998/Math/MathML”altimg = " urn: x-wiley: 03649024:媒体:jgt70057: jgt70057 -数学- 0004“威利:位置=“方程/ jgt70057 -数学- 0004. - png”祝辞& lt; mrow> & lt; mrow> & lt; mi> v< / mi> & lt; mo> & lt; / mo> & lt; mi> w< / mi> & lt; mo> \ unicode {x02208} & lt; / mo> & lt; mi> v< / mi> & lt; mrow> & lt; mo> (& lt; / mo> & lt; mi&gt G< / mi> & lt; mo>) & lt; / mo> & lt; / mrow> & lt; / mrow> & lt; / mrow> & lt; / math&gt ; ,我们给出了一个简单而精确的公式来表示在G &lt；math上随机行走的期望步数xmlns="http://www.w3.org/1998/Math/MathML" altimg=“urn:x-wiley:03649024:media:jgt70057:jgt70057-math-0005”wiley:location="equation/jgt70057-math-0005.png"><mrow><mrow>< /mrow></mrow></ mrow></mrow></ mrow></mrow></math> starting at w <math xmlns="http://www.w3.org/1998/Math/MathML" altimg=“urn:x-wiley:03649024:media:jgt70057:jgt70057-math-0006”wiley:location="equation/jgt70057-math-0006.png"><mrow><mrow>< /mrow></mrow></ mrow></mrow></ mrow></mrow></ mrow></math&gt；需要先到达v &lt；math xmlns="http://www.w3.org/1998/Math/MathML" altimg=“urn:x-wiley:03649024:media:jgt70057:jgt70057-math-0007”威利:位置= "方程/ jgt70057 -数学- 0007. png”祝辞& lt; mrow> & lt; mrow> & lt; mi> v< / mi> & lt; / mrow> & lt; / mrow> & lt; / math&gt ; .该公式仅依赖于G&lt； math xmlns="http://www.w3.org/1998/Math/MathML" altimg="urn:x-wiley:03649024:media:jgt70057:jgt70057-math-0008" wiley:location="equation/jgt70057-math-0008.png"><mrow><mrow>< /mrow></mrow></ mrow></mrow></ mrow></mrow></math&gt；。这改进并扩展了Ottolini和Steinerberger最近的结果，以及Ottolini，他考虑了这个问题对于常数和轻度消失p&lt； math xmlns="http://www.w3.org/1998/Math/MathML" altimg="urn:x-wiley:03649024:media:jgt70057:jgt70057-math-0009" wiley:location="equation/jgt70057-math-0009.png"><mrow><mrow>< /mrow></mrow></ mrow></mrow></ mrow></mrow></ mrow></math&gt；

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来源期刊

Journal of Graph Theory 数学-数学

CiteScore

1.60

自引率

22.20%

发文量

130

审稿时长

6-12 weeks

期刊介绍： The Journal of Graph Theory is devoted to a variety of topics in graph theory, such as structural results about graphs, graph algorithms with theoretical emphasis, and discrete optimization on graphs. The scope of the journal also includes related areas in combinatorics and the interaction of graph theory with other mathematical sciences. A subscription to the Journal of Graph Theory includes a subscription to the Journal of Combinatorial Designs .