Finite-size effects in the short-time height distribution of the Kardar-Parisi-Zhang equation

Abstract: We use the optimal fluctuation method to evaluate the short-time probability distribution $\mathcal{P}\left(H,L,t\right)$ of height at a single point, $H=h\left(x=0,t\right)$, of the evolving Kardar-Parisi-Zhang (KPZ) interface $h\left(x,t\right)$ on a ring of length $2L$. The process starts from a flat interface. 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 $\mathcal{P}(H)$. At large $L/\sqrt{t}$ the faster-decaying tail has a double structure: it is $L$-independent, $-\ln\mathcal{P}\sim\left|H\right|^{5/2}/t^{1/2}$, at intermediately large $|H|$, and $L$-dependent, $-\ln\mathcal{P}\sim \left|H\right|^{2}L/t$, at very large $|H|$. The transition between these two regimes is sharp and, in the large $L/\sqrt{t}$ limit, behaves as a fractional-order phase transition. The transition point $H=H_{c}^{+}$ depends on $L/\sqrt{t}$. At small $L/\sqrt{t}$, the double structure of the faster tail disappears, and only the very large-$H$ tail, $-\ln\mathcal{P}\sim \left|H\right|^{2}L/t$, is observed. The slower-decaying tail does not show any $L$-dependence at large $L/\sqrt{t}$, where it coincides with the slower tail of the GOE Tracy-Widom distribution. At small $L/\sqrt{t}$ 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 $L/\sqrt{t}$. At $L/\sqrt{t} \to 0$ the transition behaves as a mean-field-like second-order phase transition. At $|H|<|H_c^{-}|$ the slower tail behaves as $-\ln\mathcal{P}\sim \left|H\right|^{2}L/t$, whereas at $|H|>|H_c^{-}|$ it coincides with the slower tail of the GOE Tracy-Widom distribution.