Phase diagram and topological expansion in the complex quartic random matrix model

IF 3.1 1区 数学 Q1 MATHEMATICS Communications on Pure and Applied Mathematics Pub Date : 2023-09-14 DOI:10.1002/cpa.22164
Pavel Bleher, Roozbeh Gharakhloo, Kenneth T-R McLaughlin
{"title":"Phase diagram and topological expansion in the complex quartic random matrix model","authors":"Pavel Bleher,&nbsp;Roozbeh Gharakhloo,&nbsp;Kenneth T-R McLaughlin","doi":"10.1002/cpa.22164","DOIUrl":null,"url":null,"abstract":"<p>We use the Riemann–Hilbert approach, together with string and Toda equations, to study the topological expansion in the quartic random matrix model. The coefficients of the topological expansion are generating functions for the numbers <math>\n <semantics>\n <mrow>\n <msub>\n <mi>N</mi>\n <mi>j</mi>\n </msub>\n <mrow>\n <mo>(</mo>\n <mi>g</mi>\n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation>$\\mathcal {N}_j(g)$</annotation>\n </semantics></math> of 4-valent connected graphs with <i>j</i> vertices on a compact Riemann surface of genus <i>g</i>. We explicitly evaluate these numbers for Riemann surfaces of genus 0,1,2, and 3. Also, for a Riemann surface of an arbitrary genus <i>g</i>, we calculate the leading term in the asymptotics of <math>\n <semantics>\n <mrow>\n <msub>\n <mi>N</mi>\n <mi>j</mi>\n </msub>\n <mrow>\n <mo>(</mo>\n <mi>g</mi>\n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation>$\\mathcal {N}_j(g)$</annotation>\n </semantics></math> as the number of vertices tends to infinity. Using the theory of quadratic differentials, we characterize the critical contours in the complex parameter plane where phase transitions in the quartic model take place, thereby proving a result of David. These phase transitions are of the following four types: (a) one-cut to two-cut through the splitting of the cut at the origin, (b) two-cut to three-cut through the birth of a new cut at the origin, (c) one-cut to three-cut through the splitting of the cut at two symmetric points, and (d) one-cut to three-cut through the birth of two symmetric cuts.</p>","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"77 2","pages":"1405-1485"},"PeriodicalIF":3.1000,"publicationDate":"2023-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/cpa.22164","citationCount":"4","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Communications on Pure and Applied Mathematics","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/cpa.22164","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 4

Abstract

We use the Riemann–Hilbert approach, together with string and Toda equations, to study the topological expansion in the quartic random matrix model. The coefficients of the topological expansion are generating functions for the numbers N j ( g ) $\mathcal {N}_j(g)$ of 4-valent connected graphs with j vertices on a compact Riemann surface of genus g. We explicitly evaluate these numbers for Riemann surfaces of genus 0,1,2, and 3. Also, for a Riemann surface of an arbitrary genus g, we calculate the leading term in the asymptotics of N j ( g ) $\mathcal {N}_j(g)$ as the number of vertices tends to infinity. Using the theory of quadratic differentials, we characterize the critical contours in the complex parameter plane where phase transitions in the quartic model take place, thereby proving a result of David. These phase transitions are of the following four types: (a) one-cut to two-cut through the splitting of the cut at the origin, (b) two-cut to three-cut through the birth of a new cut at the origin, (c) one-cut to three-cut through the splitting of the cut at two symmetric points, and (d) one-cut to three-cut through the birth of two symmetric cuts.

Abstract Image

查看原文
分享 分享
微信好友 朋友圈 QQ好友 复制链接
本刊更多论文
复四次随机矩阵模型的相图与拓扑展开
我们利用Riemann-Hilbert方法,结合弦方程和Toda方程,研究了四次随机矩阵模型的拓扑展开。拓扑展开的系数是在g属的紧致黎曼曲面上具有j个顶点的4价连通图N j(g)$ \mathcal {N}_j(g)$的生成函数对0、1、2和3属的黎曼曲面求这些数。同样,对于任意格g的黎曼曲面,我们计算了N j(g)$ \mathcal {N}_j(g)$在顶点数趋于无穷时的渐近项。利用二次微分理论,我们描述了四次模型中发生相变的复参数平面的关键轮廓,从而证明了David的一个结果。这些相变有以下四种类型:(a)一切到二切,通过原点切割的分裂;(b)二切到三切,通过原点新切割的诞生;(c)一切到三切,通过两个对称点切割的分裂;(d)一切到三切,通过两个对称切割的诞生。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
求助全文
约1分钟内获得全文 去求助
来源期刊
CiteScore
6.70
自引率
3.30%
发文量
59
审稿时长
>12 weeks
期刊介绍: Communications on Pure and Applied Mathematics (ISSN 0010-3640) is published monthly, one volume per year, by John Wiley & Sons, Inc. © 2019. The journal primarily publishes papers originating at or solicited by the Courant Institute of Mathematical Sciences. It features recent developments in applied mathematics, mathematical physics, and mathematical analysis. The topics include partial differential equations, computer science, and applied mathematics. CPAM is devoted to mathematical contributions to the sciences; both theoretical and applied papers, of original or expository type, are included.
期刊最新文献
Issue Information - TOC First‐order Sobolev spaces, self‐similar energies and energy measures on the Sierpiński carpet On the Read‐Shockley energy for grain boundaries in 2D polycrystals Stability of perfectly matched layers for Maxwell's equations in rectangular solids Issue Information - TOC
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
现在去查看 取消
×
提示
确定
0
微信
客服QQ
Book学术公众号 扫码关注我们
反馈
×
意见反馈
请填写您的意见或建议
请填写您的手机或邮箱
已复制链接
已复制链接
快去分享给好友吧!
我知道了
×
扫码分享
扫码分享
Book学术官方微信
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术
文献互助 智能选刊 最新文献 互助须知 联系我们:info@booksci.cn
Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。
Copyright © 2023 Book学术 All rights reserved.
ghs 京公网安备 11010802042870号 京ICP备2023020795号-1