Second order cubic corrections of large deviations for perturbed random walks

We prove that the Beta random walk, introduced in [BC17] 2017, has cubic fluctuations from large deviation principle of GUE Tracy-Widom type for arbitrary values ?>0 and ?>0 parameters distribution, removing previous restrictions on their values. Furthermore, we still hold intermediate disorder regime. also show any walk space-time environment matches certain moments with As a corollary emergence trajectories ending at boundary points walks space (time-independent) i.i.d. Dirichlet dimension d=2 class asymptotic behavior parameters.