  We prove that the Beta random walk has second order cubic fluctuations from
the large deviation principle of the GUE Tracy-Widom type for arbitrary values
$\upalpha>0$ and $\upbeta>0$ of the parameters of the Beta distribution,
removing previous restrictions on their values. Furthermore, we prove that the
GUE Tracy-Widom fluctuations still hold in the intermediate disorder regime. We
also show that any random walk in space-time random environment that matches
certain moments with the Beta random walk also has GUE Tracy-Widom fluctuations
in the intermediate disorder regime. As a corollary we show the emergence of
GUE Tracy-Widom fluctuations from the large deviation principle for
trajectories ending at boundary points for random walks in space
(time-independent) i.i.d. Dirichlet random environment in dimension $d=2$ for a
class of asymptotic behavior of the parameters.
