{"title":"Excessive symmetry can preclude cutoff","authors":"Eric Ramos , Graham White","doi":"10.1016/j.laa.2024.06.025","DOIUrl":null,"url":null,"abstract":"<div><p>In this paper we look at the families of random walks arising from FI-graphs. One may think of these objects as families of nested graphs, each equipped with a natural action by a symmetric group <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, such that these actions are compatible and transitive. Families of graphs of this form were introduced by the authors in <span>[9]</span>, while a systematic study of random walks on these families were considered in <span>[10]</span>. In the present work, we illustrate that these random walks never exhibit the so-called product condition, and therefore also never display total variation cutoff as defined by Aldous and Diaconis <span>[1]</span>. In particular, we provide a large family of algebro-combinatorially motivated examples of collections of Markov chains which satisfy some well-known algebraic heuristics for cutoff, while not actually having the property.</p></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":null,"pages":null},"PeriodicalIF":1.0000,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Linear Algebra and its Applications","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0024379524002799","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
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Abstract
In this paper we look at the families of random walks arising from FI-graphs. One may think of these objects as families of nested graphs, each equipped with a natural action by a symmetric group , such that these actions are compatible and transitive. Families of graphs of this form were introduced by the authors in [9], while a systematic study of random walks on these families were considered in [10]. In the present work, we illustrate that these random walks never exhibit the so-called product condition, and therefore also never display total variation cutoff as defined by Aldous and Diaconis [1]. In particular, we provide a large family of algebro-combinatorially motivated examples of collections of Markov chains which satisfy some well-known algebraic heuristics for cutoff, while not actually having the property.
期刊介绍：
Linear Algebra and its Applications publishes articles that contribute new information or new insights to matrix theory and finite dimensional linear algebra in their algebraic, arithmetic, combinatorial, geometric, or numerical aspects. It also publishes articles that give significant applications of matrix theory or linear algebra to other branches of mathematics and to other sciences. Articles that provide new information or perspectives on the historical development of matrix theory and linear algebra are also welcome. Expository articles which can serve as an introduction to a subject for workers in related areas and which bring one to the frontiers of research are encouraged. Reviews of books are published occasionally as are conference reports that provide an historical record of major meetings on matrix theory and linear algebra.
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