{"title":"Monotonicity properties for Bernoulli percolation on layered graphs— A Markov chain approach","authors":"Philipp König, Thomas Richthammer","doi":"10.1016/j.spa.2024.104549","DOIUrl":null,"url":null,"abstract":"<div><div>A layered graph <span><math><msup><mrow><mi>G</mi></mrow><mrow><mo>×</mo></mrow></msup></math></span> is the Cartesian product of a graph <span><math><mrow><mi>G</mi><mo>=</mo><mrow><mo>(</mo><mi>V</mi><mo>,</mo><mi>E</mi><mo>)</mo></mrow></mrow></math></span> with the linear graph <span><math><mi>Z</mi></math></span>, e.g. <span><math><msup><mrow><mi>Z</mi></mrow><mrow><mo>×</mo></mrow></msup></math></span> is the 2D square lattice <span><math><msup><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>. For Bernoulli percolation with parameter <span><math><mrow><mi>p</mi><mo>∈</mo><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></mrow></math></span> on <span><math><msup><mrow><mi>G</mi></mrow><mrow><mo>×</mo></mrow></msup></math></span> one intuitively would expect that <span><math><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>p</mi></mrow></msub><mrow><mo>(</mo><mrow><mo>(</mo><mi>o</mi><mo>,</mo><mn>0</mn><mo>)</mo></mrow><mo>↔</mo><mrow><mo>(</mo><mi>v</mi><mo>,</mo><mi>n</mi><mo>)</mo></mrow><mo>)</mo></mrow><mo>≥</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>p</mi></mrow></msub><mrow><mo>(</mo><mrow><mo>(</mo><mi>o</mi><mo>,</mo><mn>0</mn><mo>)</mo></mrow><mo>↔</mo><mrow><mo>(</mo><mi>v</mi><mo>,</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mo>)</mo></mrow></mrow></math></span> for all <span><math><mrow><mi>o</mi><mo>,</mo><mi>v</mi><mo>∈</mo><mi>V</mi></mrow></math></span> and <span><math><mrow><mi>n</mi><mo>≥</mo><mn>0</mn></mrow></math></span>. This is reminiscent of the better known bunkbed conjecture. Here we introduce an approach to the above monotonicity conjecture that makes use of a Markov chain building the percolation pattern layer by layer. In case of finite <span><math><mi>G</mi></math></span> we thus can show that for some <span><math><mrow><mi>N</mi><mo>≥</mo><mn>0</mn></mrow></math></span> the above holds for all <span><math><mrow><mi>n</mi><mo>≥</mo><mi>N</mi></mrow></math></span> <span><math><mrow><mi>o</mi><mo>,</mo><mi>v</mi><mo>∈</mo><mi>V</mi></mrow></math></span> and <span><math><mrow><mi>p</mi><mo>∈</mo><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></mrow></math></span>. One might hope that this Markov chain approach could be useful for other problems concerning Bernoulli percolation on layered graphs.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"181 ","pages":"Article 104549"},"PeriodicalIF":1.1000,"publicationDate":"2024-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Stochastic Processes and their Applications","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0304414924002576","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"STATISTICS & PROBABILITY","Score":null,"Total":0}
引用次数: 0
Abstract
A layered graph is the Cartesian product of a graph with the linear graph , e.g. is the 2D square lattice . For Bernoulli percolation with parameter on one intuitively would expect that for all and . This is reminiscent of the better known bunkbed conjecture. Here we introduce an approach to the above monotonicity conjecture that makes use of a Markov chain building the percolation pattern layer by layer. In case of finite we thus can show that for some the above holds for all and . One might hope that this Markov chain approach could be useful for other problems concerning Bernoulli percolation on layered graphs.
期刊介绍:
Stochastic Processes and their Applications publishes papers on the theory and applications of stochastic processes. It is concerned with concepts and techniques, and is oriented towards a broad spectrum of mathematical, scientific and engineering interests.
Characterization, structural properties, inference and control of stochastic processes are covered. The journal is exacting and scholarly in its standards. Every effort is made to promote innovation, vitality, and communication between disciplines. All papers are refereed.