The celebrated Kardar-Parisi-Zhang (KPZ) equation is a standard model of non-equilibrium stochastic growth. In the recent years the focus of studies of the KPZ equation and related systems shifted towards detailed quantities, such as the full distribution P(H,t) of the interface height at a point, h(0,t)-h(0,0)=H. Refs. [1-3] studied the short-time height distribution of the KPZ interface in an infinite one-dimensional system starting from a flat interface. These results were obtained using the optimal fluctuation method (OFM).

In this work we study finite-size effects in this system [4] by considering a ring of length 2L . At short times typical (small) height fluctuations are unaffected by the KPZ nonlinearity and belong to the Edwards-Wilkinson universality class. The nonlinearity, however, strongly affects the (asymmetric) tails of P(H). At large L/√t the faster-decaying tail has a double structure: it is L-independent, -lnP ~ |H|^5/2/t^1/2, at intermediately large |H|, and L-dependent, -lnP ~ |H|²L/t, at very large |H|. The transition between these two regimes is sharp and, in the large L/√t limit, behaves as a fractional-order phase transition. The transition point H=H_c⁺ depends on L/√t. At small L/√t, the double structure of the faster tail disappears, and only the very large-H tail, -lnP ~ |H|²L/t, is observed. The slower-decaying tail does not show any L-dependence at large L/√t, where it coincides with the slower tail of the GOE Tracy-Widom distribution. At small L/√t this tail also has a double structure. The transition between the two regimes occurs at a value of height H=H_c^- which depends on L/√t. At L/√t→0 the transition behaves as a mean-field-like second-order phase transition. At |H|<|H_c^-| the slower tail behaves as -lnP ~ |H|²L/t, whereas at |H|>|H_c^-| it coincides with the slower tail of the GOE Tracy-Widom distribution.

[1] I.V. Kolokolov and S.E. Korshunov, Phys. Rev. B 75, 140201(R) (2007).

[2] I.V. Kolokolov and S.E. Korshunov, Phys. Rev. E 80, 031107 (2009).

[3] B. Meerson, E. Katzav, and A. Vilenkin, Phys. Rev. Lett. 116, 070601 (2016).

[4] N. R. Smith, B. Meerson and P. V. Sasorov, arXiv:1710.04188.