The limiting distribution of Legendre paths

Let $p$ be a prime number and $\left(\frac{\cdot }{p}\right)$ be the Legendre symbol modulo $p$. The \emph{Legendre path} attached to $p$ is the polygonal path whose vertices are the normalized character sums $\frac{1}{\sqrt{p}}{\sum }_{n\le j}\left(\frac{n}{p}\right)$ for $0\le j\le p-1$. In this paper, we investigate the distribution of Legendre paths as we vary over the primes $Q\le p\le 2Q$, when $Q$ is large. Our main result shows that as $Q\to \mathrm{\infty }$, these paths converge in law, in the space of real-valued continuous functions on $[0,1]$, to a certain random Fourier series constructed using Rademacher random completely multiplicative functions. This was previously proved by the first author under the assumption of the Generalized Riemann Hypothesis.