Asymptotic integral kernel for ensembles of random normal matrices with radial potentials
Asymptotic integral kernel for ensembles of random normal matrices with radial potentials
The method of steepest descent is used to study the integral kernel of a family of normal random matrix ensembles with eigenvalue distribution \documentclass[12pt]{minimal}\begin{document}$P_{N}(z_{1},\cdots ,z_{N})=Z_{N}^{-1}\emph {e}^{-N\sum _{i=1}^{N}V_{\alpha }(z_{i})}\prod _{1\le i<j\le N}\left|z_{i}-z_{j}\right|^{2},$\end{document}PN(z1,⋯,zN)=ZN−1e−N∑i=1NVα(zi)∏1≤i<j≤Nzi−zj2,where Vα(z) = |z|α, \documentclass[12pt]{minimal}\begin{document}$z\in \mathbb {C}$\end{document}z∈C and α ∈ ]0, ∞[. Asymptotic formulas with error estimate on sectors are …