A famous result going back to Eric Kostlan states that the moduli of the eigenvalues of random normal matrices with radial potential are independent yet non-identically distributed. This phenomenon is at the heart of the asymptotic analysis of the edge, and leads in particular to the Gumbel fluctuation of the spectral radius when the potential is quadratic. In the present work, we show that a wide variety of laws of fluctuation are possible, beyond the already known cases, including for instance Gumbel and exponential laws at unusual speeds. We study the convergence in law of the spectral radius as well as the limiting point process at the edge. Our work can also be seen as the asymptotic analysis of the edge of two-dimensional determinantal Coulomb gases and the identification of the limiting kernels.

追溯到 Eric Kostlan 的一个著名结果指出，具有径向势的随机正态矩阵的特征值的模是独立的但不均匀分布。这种现象是边缘渐近分析的核心，特别是当势为二次时会导致光谱半径的 Gumbel 波动。在目前的工作中，我们表明，除了已知的情况之外，各种波动定律都是可能的，包括例如 Gumbel 和以异常速度的指数定律。我们研究了光谱半径的收敛规律以及边缘的极限点过程。我们的工作也可以看作是对二维行列式库仑气体边缘的渐近分析和限制核的识别。