  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 $\frac{1}{\sqrt{p}}
\sum_{n\leq j} \left(\frac{n}{p}\right)$ for $0\leq j\leq p-1$. In this paper,
we investigate the distribution of Legendre paths as we vary over the primes
$Q\leq p\leq 2Q$, when $Q$ is large. Our main result shows that as $Q \to
\infty$, 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.
