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

We use the optimal fluctuation method to evaluate the short-time probability distribution $P(H,L,t)$ of height at a single point, $H=h(x=0,t)$, of the evolving Kardar–Parisi–Zhang (KPZ) interface $h(x,t)$ 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 $P\left(H\right)$. At large $L/\sqrt{t}$ the faster-decaying tail has a double structure: it is L-independent, $-\ln P\sim {\left|H\right|}^{5/2}/t^{1/2}$, at intermediately large $\left|H\right|$, and L-dependent, $-\ln P\sim {\left|H\right|}^{2}L/t$, at very large $\left|H\right|$. 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, $-\ln P\sim {\left|H\right|}^{2}L/t$, 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 $\left|H\right|<\left|H_c^{-}\right|$ the slower tail behaves as $-\ln P\sim {\left|H\right|}^{2}L/t$, whereas at $\left|H\right|>\left|H_c^{-}\right|$ it coincides with the slower tail of the GOE Tracy–Widom distribution. (paper: classical statistical mechanics, equilibrium and non-equilibrium)