Random Riesz energies on compact Kähler manifolds
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 be viewed as polynomials in \(m\) variables of degree \(N\)), the coefficients \(a_j\) are null.NEWLINENEWLINEThese results generalize the work by \textit{Q. Zhong} [Indiana Univ. Math. J. 57, No. 4, 1753--1780 (2008; Zbl 1175.30040)] for \(M\) a Riemann surface and the interaction energy given by the Green kernel \(G_\sigma(p,q)\) whose singularity on the diagonal is similar to that of the Riesz energy. Zhong proves that the average energy of such random zeros is of the same order of magnitude as that of minimal energy configurations for large \(N\), a similar result has been proved for the Riesz logarithmic energy on the \(2\)-sphere by \textit{D. Armentano} et al. [Trans. Am. Math. Soc. 363, No. 6, 2955--2965 (2011; Zbl 1223.31003)]; these results are a strong motivation to study the expected energy, even if the description of minimal energy configurations and their energy remain unsolved. The authors discuss how their asymptotics give indications on the local behaviour (repulsion/attraction) of the zero points configurations and the long-range order (uniform distribution).NEWLINENEWLINEThe proof for this asymptotics is based on the analysis for large \(N\) of the diagonal of the Bergman kernel, where the Bergman operator is the projection \(\Pi_N(p,q):L^2(M,L^N)\to H^0(M,L^N)\) onto the holomorphic sections.NEWLINENEWLINEFor \(k<m\), submanifolds determined as zeros of random \(L^N\)-sections families \(\mathbf{S}_k=(s_1,\ldots, s_k)\) can be given an energy; expected energy asymptotics for large \(N\) similar to those in the discrete case are also proved.
scientific article; zbMATH DE number 6205216
scientific article