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We prove that the Beta random walk has second order cubic fluctuations fromthe 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 theGUE Tracy-Widom fluctuations still hold in the intermediate disorder regime. Wealso show that any random walk in space-time random environment that matchescertain moments with the Beta random walk also has GUE Tracy-Widom fluctuationsin the intermediate disorder regime. As a corollary we show the emergence ofGUE Tracy-Widom fluctuations from the large deviation principle fortrajectories ending at boundary points for random walks in space(time-independent) i.i.d. Dirichlet random environment in dimension $d=2$ for aclass of asymptotic behavior of the parameters.

Second order cubic corrections of large deviations for perturbed random walks