A universal spacing law hidden in random math holds true no matter what shape you draw it on.
Random zeros in Gaussian analytic functions were thought to be chaotic and dependent on the underlying geometry. This discovery shows that the distances between these zeros actually converge to a universal Poisson process. This law remains the same whether the math is done on a flat plane or a complex Riemann surface. It suggests that there is a deep, fundamental order underlying what appears to be pure randomness. This insight could help scientists better understand the distribution of everything from energy levels to social networks. Hidden structure exists in the most unlikely places in the mathematical world.
Smallest distances between zeros of Gaussian analytic functions
arXiv · 2604.26316
In this article, we study the smallest distances between the zeros of Gaussian analytic functions over compact Riemann surfaces. Our main result is that, after appropriate rescaling, the point process of the smallest distances converge to a Poisson point process with a universal rate. Furthermore, the locations where these smallest distances occur tend to follow a uniform measure with respect to the volume form. As a consequence, the limiting density of the $k$-th rescaled smallest distance is p