GOE fluctuations for the maximum of the top path in alternating sign matrices

<!doctype html> <html lang="en"> <head> <meta charset="utf-8"> <meta http-equiv="x-ua-compatible" content="ie=edge"> <meta name="viewport" content="width=device-width, initial-scale=1.0"> <meta name="theme-color" content="#8e44ad"> <meta name="description" content="The six-vertex model is an important toy-model in statistical mechanics for two-dimensional ice with a natural parameter ∆. When ∆ = 0, the so-called free-..." /> <meta name="robots" content="noarchive">  <meta name="citation_title" content="GOE fluctuations for the maximum of the top path in alternating sign matrices"> <meta name="citation_abstract" lang="en" content="The six-vertex model is an important toy-model in statistical mechanics for two-dimensional ice with a natural parameter ∆. When ∆ = 0, the so-called free-fermion point, the model is in natural correspondence with domino tilings of the Aztec diamond. Although this model is integrable for all ∆, there has been very little progress in understanding its statistics in the scaling limit for other values. In this work, we focus on the six-vertex model with domain wall boundary conditions at ∆ = 1/2, where it corresponds to alternating sign matrices (ASMs). We consider the level lines in a height function representation of ASMs. We show that the maximum of the topmost level line for a uniformly random ASMs has the GOE Tracy–Widom distribution after appropriate rescaling. A key ingredient in our proof is Zeilberger’s proof of the ASM conjecture. As far as we know, this is the first edge fluctuation result away from the tangency points for the domain-wall six-vertex model when we are not in the free fermion case."> <meta name="citation_author" content="Ayyer, Arvind"> <meta name="citation_author" content="Chhita, Sunil"> <meta name="citation_author" content="Johansson, Kurt"> <meta name="citation_publication_date" content="2023/07/15"> <meta name="citation_journal_title" content="Duke Mathematical Journal"> <meta name="citation_volume" content="172"> <meta name="citation_issue" content="10"> <meta name="citation_doi" content="10.1215/00127094-2022-0075"> <meta name="citation_publisher" content="Duke University Press"> <meta name="citation_issn" content="0012-7094" /> <meta name="citation_issn" content="1547-7398" /> <meta name="citation_public_url" content="https://durham-repository.worktribe.com/output/1201674/goe-fluctuations-for-the-maximum-of-the-top-path-in-alternating-sign-matrices" /> <meta name="citation_abstract_html_url" content="https://durham-repository.worktribe.com/output/1201674/goe-fluctuations-for-the-maximum-of-the-top-path-in-alternating-sign-matrices.abstract" /> <meta name="citation_pdf_url" content="https://durham-repository.worktribe.com/preview/1201700/36419.pdf"> <meta name="citation_article_type" content="Journal Article" />  <title>GOE fluctuations for the maximum of the top path in alternating sign matrices</title> <link rel="stylesheet" href="/css/surface.min.css?a=a" /> <link rel="stylesheet" href="/css/app.css?a=d" /> <link rel="stylesheet" href="/font-awesome-4.7.0/css/font-awesome.min.css"> <link href="/css/irus_r5_widget.css" rel="stylesheet"> <style> #irus_widget_container { font-family: "Muli","Helvetica Neue",Arial,sans-serif; } #irus_widget_form { background-color: #f9f5fb; padding: 2em; } #irus_widget_form > div { margin-bottom: 5px; } </style> <link rel="stylesheet" href="/css/skins/Durham.css?a=b" /> <link 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Preview" style="cursor: pointer;" onclick="$('[data-preview]').click();"> <img style="box-shadow: 0 1px 3px 0 rgba(0, 0, 0, 0.1), 0 1px 2px 0 rgba(0, 0, 0, 0.06);" src="https://durham-repository.worktribe.com/thumbnail/1201674" alt="GOE fluctuations for the maximum of the top path in alternating sign matrices Thumbnail" > </figure> <br /><br /> <h2 class="subtitle">Authors</h2> <div style="max-height: 100vh; overflow: auto;"> <article class="media"> <div class="media-content"> <div class="content"> <p> <strong>Arvind Ayyer</strong> </p> </div> </div> </article> <article class="media"> <figure class="media-left"> <p class="image is-64x64"> <img src="https://durham-repository.worktribe.com/avatar/209870" alt="Profile image of Sunil Chhita"> </p> </figure> <div class="media-content"> <div class="content"> <p> <a href="https://durham-repository.worktribe.com/person/209870/sunil-chhita"><strong>Professor Sunil Chhita</strong></a> <small>sunil.chhita@durham.ac.uk</small> <br> Early Career Fellowship </p> </div> </div> <div class="media-right"> <a href="https://durham-repository.worktribe.com/person/209870/sunil-chhita" class="level-right" aria-label="Sunil Chhita"> <span class="icon is-small level-right"><i class="fa fa-info-circle" aria-hidden="true"></i></span> </a> </div> </article> <article class="media"> <div class="media-content"> <div class="content"> <p> <strong>Kurt Johansson</strong> </p> </div> </div> </article> </div> <br /><br /> </div> <div class="column"> <div class="content abstract"> <h3 class="subtitle">Abstract</h3> <p class="description">The six-vertex model is an important toy-model in statistical mechanics for two-dimensional ice with a natural parameter ∆. When ∆ = 0, the so-called free-fermion point, the model is in natural correspondence with domino tilings of the Aztec diamond. Although this model is integrable for all ∆, there has been very little progress in understanding its statistics in the scaling limit for other values. In this work, we focus on the six-vertex model with domain wall boundary conditions at ∆ = 1/2, where it corresponds to alternating sign matrices (ASMs). We consider the level lines in a height function representation of ASMs. We show that the maximum of the topmost level line for a uniformly random ASMs has the GOE Tracy–Widom distribution after appropriate rescaling. A key ingredient in our proof is Zeilberger’s proof of the ASM conjecture. As far as we know, this is the first edge fluctuation result away from the tangency points for the domain-wall six-vertex model when we are not in the free fermion case.</p> </div> <div class="content citation"> <h3 class="subtitle">Citation</h3> <p class="description"> Ayyer, A., Chhita, S., & Johansson, K. (2023). GOE fluctuations for the maximum of the top path in alternating sign matrices. Duke Mathematical Journal, 172(10), 1961-2104. <a style="text-decoration: underline;" href="https://doi.org/10.1215/00127094-2022-0075" target="_blank">https://doi.org/10.1215/00127094-2022-0075</a> </p> </div> <table class="table" style="word-break: break-word; width:100%;"> <tbody> <tr> <th class="table-background" width="35%">Journal Article Type</th> <td>Article</td> </tr> <tr> <th class="table-background" width="35%">Acceptance Date</th> <td>Jul 6, 2022</td> </tr> <tr> <th class="table-background" width="35%">Online Publication Date</th> <td>Aug 15, 2023</td> </tr> <tr> <th class="table-background" width="35%">Publication Date</th> <td>Jul 15, 2023</td> </tr> <tr> <th class="table-background" width="35%">Deposit Date</th> <td>Jul 7, 2022</td> </tr> <tr> <th class="table-background" width="35%">Publicly Available Date</th> <td>Jul 15, 2023</td> </tr> <tr> <th class="table-background" width="35%">Journal</th> <td>Duke Mathematical Journal</td> </tr> <tr> <th class="table-background" width="35%">Print ISSN</th> <td>0012-7094</td> </tr> <tr> <th class="table-background" width="35%">Electronic ISSN</th> <td>1547-7398</td> </tr> <tr> <th class="table-background" width="35%">Publisher</th> <td>Duke University Press</td> </tr> <tr> <th class="table-background" width="35%">Peer Reviewed</th> <td>Peer Reviewed</td> </tr> <tr> <th class="table-background" width="35%">Volume</th> <td>172</td> </tr> <tr> <th class="table-background" width="35%">Issue</th> <td>10</td> </tr> <tr> <th class="table-background" width="35%">Pages</th> <td>1961-2104</td> </tr> <tr> <th class="table-background" width="35%">DOI</th> <td><a style="text-decoration: underline;" href="https://doi.org/10.1215/00127094-2022-0075" target="_blank">https://doi.org/10.1215/00127094-2022-0075</a></td> </tr> <tr> <th class="table-background" width="35%">Public URL</th> <td><a style="text-decoration: underline;" href="https://durham-repository.worktribe.com/output/1201674" 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GOE fluctuations for the maximum of the top path in alternating sign matrices