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.
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