Random Riesz energies on compact Kähler manifolds
type: Item
instance of: scholarly article
title: Random Riesz energies on compact Kähler manifolds
zbMATH Open document ID: 1295.32013
DOI: 10.1090/S0002-9947-2013-05870-9
author: Renjie Feng
author: Steve Zelditch
published in: Transactions of the American Mathematical Society
publication date: 2013-09-04T00:00:00Z
full work available at URL: https://arxiv.org/abs/1112.3993
Mathematics Subject Classification ID: 32A60
Mathematics Subject Classification ID: 32L10
Mathematics Subject Classification ID: 60D05
Mathematics Subject Classification ID: 32A25
zbMATH DE Number: 6205216
zbMATH Keywords: positive holomophic line bundles
zbMATH Keywords: closed Kähler manifold
zbMATH Keywords: random sections
zbMATH Keywords: interaction energy
zbMATH Keywords: Riesz energy
zbMATH Keywords: Bergman kernel
MaRDI profile type: Publication
arXiv ID: 1112.3993
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review text: Let \((L,h)\to(M,g) \) be a Hermitian positive holomorphic line bundle over an \(m\)-dimensional closed Kähler manifold where the Hermitian metric \(h\) is determined by the Kähler potential \(\phi\) through \(h=e^{-\phi}\). Let \(L^N\) be the \(N\)th tensor product \(L^{\otimes N}\to M\) with \(H^0(M,L^N)\) its global holomorphic sections space (which is finite dimensional). The natural scalar product on \(H^0(M,L^N)\) induces Gaussian measures \(\otimes^k\mu_N\) (shortly denoted \(\mu_N\)) on the space \(H^0(M,L^N)^k\), whose elements \(\mathbf{S}_k=(s_1,\ldots, s_k)\in H^0(M,L^N)^k\) are independent Gaussian random sections. For a family \(\mathbf{S}_m\), its zero set \(Z({\mathbf{S}_m})=\{s_1=\ldots=s_m=0\}\) is \(\mu_N\)-almost surely discrete (hence finite) and its energy \({\mathcal E}({\mathbf{S}_m})\) is defined as the sum of all pair interaction energies NEWLINENEWLINENEWLINE\[NEWLINE{\mathcal E}(\mathbf{S}_m)=\sum_{p\not=q\in Z(\text\textbf{S}_m)}{\mathcal E}(p,q).NEWLINE\]NEWLINE The authors consider the \(\sigma\)-Riesz interaction energy \({\mathcal E}_\sigma\) induced by the Riemannian distance \(d_g\) on \(M\): if \(\sigma>0\), \({\mathcal E}_\sigma(p,q)=d_g(p,q)^{-\sigma}\) and \({\mathcal E}_0(p,q)=-\log d_g(p,q)\). NEWLINENEWLINENEWLINENEWLINE The main statement of the paper is the asymptotic expansion for \(N\to\infty\) of the expected Riesz energy along the Gaussian measure \(\mu_N\) when \(\sigma\in[0,\min(2m,4))\): if \(\sigma >0\), the expansion is NEWLINE\[NEWLINE \mathbb E_{\mu_N}({\mathcal E}_\sigma) =c_1N^{2m}+\sum_{j=2}^{[m-\sigma]}a_jN^{2m-j}+c_mN^{m+\sigma/2}+O\big(N^{m+(\sigma-1)/2}(\log N)^{m-\sigma/2}\big). NEWLINE\]NEWLINE The coefficients \(a_j, c_m\) are the integrals of the interaction kernel \({\mathcal E}_\sigma(p,q)\) against the curvature invariants of the metric \(h\). In the case of the projective space \(\mathbb C\mathbb P^m\) with its canonical line bundle \(L={\mathcal O}(1)\) (so the sections of \({\mathcal O}(N)\) can…
reviewed by: Laurent Guillopé
description: scientific article; zbMATH DE number 6205216