{"title":"随机正规矩阵模型中的平面正交多项式与边界普适性","authors":"H. Hedenmalm, Aron Wennman","doi":"10.4310/acta.2021.v227.n2.a3","DOIUrl":null,"url":null,"abstract":"We prove that the planar normalized orthogonal polynomials $P_{n,m}(z)$ of degree $n$ with respect to an exponentially varying planar measure $e^{-2mQ(z)}\\mathrm{dA}(z)$ enjoy an asymptotic expansion \\[ P_{n,m}(z)\\sim m^{\\frac{1}{4}}\\sqrt{\\phi_\\tau'(z)}[\\phi_\\tau(z)]^n e^{m\\mathcal{Q}_\\tau(z)}\\left(\\mathcal{B}_{0,\\tau}(z) +\\frac{1}{m}\\mathcal{B}_{1,\\tau}(z)+\\frac{1}{m^2} \\mathcal{B}_{2,\\tau}(z)+\\ldots\\right), \\] as $n=m\\tau\\to\\infty$. Here $\\mathcal{S}_\\tau$ denotes the droplet, the boundary of which is assumed to be a smooth, simple, closed curve, and $\\phi_\\tau$ is a conformal mapping $\\mathcal{S}_\\tau^c\\to\\mathbb{D}_e$. The functions $\\mathcal{Q}_\\tau$ and $\\mathcal{B}_{j,\\tau}(z)$ are bounded holomorphic functions which may be computed in terms of $Q$ and $\\mathcal{S}_\\tau$. We apply these results to prove universality at the boundary for regular droplets in the random normal matrix model, i.e., that the limiting rescaled process is the random process with correlation kernel $$ \\mathrm{k}(\\xi,\\eta)= e^{\\xi\\bar\\eta\\,-\\frac12(\\lvert\\xi\\rvert^2+\\lvert \\eta\\rvert^2)} \\operatorname{erf}(\\xi+\\bar{\\eta}). $$ A key ingredient in the proof of the asymptotic expansion is the construction of an {orthogonal foliation} -- a smooth flow of closed curves near $\\partial\\mathcal{S}_\\tau$, on each of which $P_{n,m}$ is orthogonal to lower order polynomials, with respect to an induced measure. To compute the coefficients, we develop an algorithm which determines $\\mathcal{B}_{j,\\tau}$ up to any desired order in terms of inhomogeneous Toeplitz kernel conditions.","PeriodicalId":4,"journal":{"name":"ACS Applied Energy Materials","volume":null,"pages":null},"PeriodicalIF":5.4000,"publicationDate":"2017-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"43","resultStr":"{\"title\":\"Planar orthogonal polynomials and boundary universality in the random normal matrix model\",\"authors\":\"H. Hedenmalm, Aron Wennman\",\"doi\":\"10.4310/acta.2021.v227.n2.a3\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We prove that the planar normalized orthogonal polynomials $P_{n,m}(z)$ of degree $n$ with respect to an exponentially varying planar measure $e^{-2mQ(z)}\\\\mathrm{dA}(z)$ enjoy an asymptotic expansion \\\\[ P_{n,m}(z)\\\\sim m^{\\\\frac{1}{4}}\\\\sqrt{\\\\phi_\\\\tau'(z)}[\\\\phi_\\\\tau(z)]^n e^{m\\\\mathcal{Q}_\\\\tau(z)}\\\\left(\\\\mathcal{B}_{0,\\\\tau}(z) +\\\\frac{1}{m}\\\\mathcal{B}_{1,\\\\tau}(z)+\\\\frac{1}{m^2} \\\\mathcal{B}_{2,\\\\tau}(z)+\\\\ldots\\\\right), \\\\] as $n=m\\\\tau\\\\to\\\\infty$. Here $\\\\mathcal{S}_\\\\tau$ denotes the droplet, the boundary of which is assumed to be a smooth, simple, closed curve, and $\\\\phi_\\\\tau$ is a conformal mapping $\\\\mathcal{S}_\\\\tau^c\\\\to\\\\mathbb{D}_e$. The functions $\\\\mathcal{Q}_\\\\tau$ and $\\\\mathcal{B}_{j,\\\\tau}(z)$ are bounded holomorphic functions which may be computed in terms of $Q$ and $\\\\mathcal{S}_\\\\tau$. We apply these results to prove universality at the boundary for regular droplets in the random normal matrix model, i.e., that the limiting rescaled process is the random process with correlation kernel $$ \\\\mathrm{k}(\\\\xi,\\\\eta)= e^{\\\\xi\\\\bar\\\\eta\\\\,-\\\\frac12(\\\\lvert\\\\xi\\\\rvert^2+\\\\lvert \\\\eta\\\\rvert^2)} \\\\operatorname{erf}(\\\\xi+\\\\bar{\\\\eta}). $$ A key ingredient in the proof of the asymptotic expansion is the construction of an {orthogonal foliation} -- a smooth flow of closed curves near $\\\\partial\\\\mathcal{S}_\\\\tau$, on each of which $P_{n,m}$ is orthogonal to lower order polynomials, with respect to an induced measure. To compute the coefficients, we develop an algorithm which determines $\\\\mathcal{B}_{j,\\\\tau}$ up to any desired order in terms of inhomogeneous Toeplitz kernel conditions.\",\"PeriodicalId\":4,\"journal\":{\"name\":\"ACS Applied Energy Materials\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":5.4000,\"publicationDate\":\"2017-10-17\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"43\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"ACS Applied Energy Materials\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.4310/acta.2021.v227.n2.a3\",\"RegionNum\":3,\"RegionCategory\":\"材料科学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"CHEMISTRY, PHYSICAL\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"ACS Applied Energy Materials","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.4310/acta.2021.v227.n2.a3","RegionNum":3,"RegionCategory":"材料科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"CHEMISTRY, PHYSICAL","Score":null,"Total":0}
Planar orthogonal polynomials and boundary universality in the random normal matrix model
We prove that the planar normalized orthogonal polynomials $P_{n,m}(z)$ of degree $n$ with respect to an exponentially varying planar measure $e^{-2mQ(z)}\mathrm{dA}(z)$ enjoy an asymptotic expansion \[ P_{n,m}(z)\sim m^{\frac{1}{4}}\sqrt{\phi_\tau'(z)}[\phi_\tau(z)]^n e^{m\mathcal{Q}_\tau(z)}\left(\mathcal{B}_{0,\tau}(z) +\frac{1}{m}\mathcal{B}_{1,\tau}(z)+\frac{1}{m^2} \mathcal{B}_{2,\tau}(z)+\ldots\right), \] as $n=m\tau\to\infty$. Here $\mathcal{S}_\tau$ denotes the droplet, the boundary of which is assumed to be a smooth, simple, closed curve, and $\phi_\tau$ is a conformal mapping $\mathcal{S}_\tau^c\to\mathbb{D}_e$. The functions $\mathcal{Q}_\tau$ and $\mathcal{B}_{j,\tau}(z)$ are bounded holomorphic functions which may be computed in terms of $Q$ and $\mathcal{S}_\tau$. We apply these results to prove universality at the boundary for regular droplets in the random normal matrix model, i.e., that the limiting rescaled process is the random process with correlation kernel $$ \mathrm{k}(\xi,\eta)= e^{\xi\bar\eta\,-\frac12(\lvert\xi\rvert^2+\lvert \eta\rvert^2)} \operatorname{erf}(\xi+\bar{\eta}). $$ A key ingredient in the proof of the asymptotic expansion is the construction of an {orthogonal foliation} -- a smooth flow of closed curves near $\partial\mathcal{S}_\tau$, on each of which $P_{n,m}$ is orthogonal to lower order polynomials, with respect to an induced measure. To compute the coefficients, we develop an algorithm which determines $\mathcal{B}_{j,\tau}$ up to any desired order in terms of inhomogeneous Toeplitz kernel conditions.
期刊介绍:
ACS Applied Energy Materials is an interdisciplinary journal publishing original research covering all aspects of materials, engineering, chemistry, physics and biology relevant to energy conversion and storage. The journal is devoted to reports of new and original experimental and theoretical research of an applied nature that integrate knowledge in the areas of materials, engineering, physics, bioscience, and chemistry into important energy applications.