Poisson approximation of the length spectrum of random surfaces

TL;DR

This paper applies the Chen-Stein method to approximate the length spectrum of random surfaces with a multivariate Poisson distribution, providing explicit error bounds and establishing the approximation's validity for certain curve lengths.

Contribution

It introduces a novel application of the Chen-Stein method to the length spectrum of random surfaces, with explicit error bounds and a range of applicability.

Findings

Poisson approximation applies to curves of length up to o(log log g)

Explicit error bounds are derived for the approximation

The approach simplifies analysis of the length spectrum in random surfaces

Abstract

Multivariate Poisson approximation of the length spectrum of random surfaces is studied by means of the Chen-Stein method. This approach delivers simple and explicit error bounds in Poisson limit theorems. They are used to prove that Poisson approximation applies to curves of length up to order $o(\log\log g)$ with $g$ being the genus of the surface.