# The shape of quadratic Gauss paths

## TL;DR

This paper studies the distribution of quadratic Gauss paths, showing they converge to a specific random Fourier series and providing a classification of their limiting shapes, explaining their notable visual features.

## Contribution

It introduces a new probabilistic description of quadratic Gauss paths and characterizes their limiting behavior as the parameter grows large.

## Key findings

- Quadratic Gauss paths converge in law to a random Fourier series.
- The paper provides a classification of the limiting shapes of these paths.
- It establishes convergence in probability for the ensemble of paths.

## Abstract

We consider the distribution of quadratic Gauss paths, polygonal paths joining partial sums of quadratic Gauss sums to square-free fundamental discriminant moduli in a dyadic range [Q,2Q]. We prove that this striking ensemble converges in law, as Q->\infty, to a random Fourier series we explicitly describe, and we prove a convergence in probability result and a classification result for the limiting shapes that explain the visually remarkable properties of these Gauss paths.