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The dynamical regimes of models belonging to the Kardar-Parisi-Zhang (KPZ)universality class are investigated in d=2+1 by extensive simulationsconsidering flat and curved geometries. Geometry-dependent universaldistributions, different from their Tracy-Widom counterpart in one-dimension,were found. Distributions exhibit finite-time corrections hallmarked by a shiftin the mean decaying as t^-\beta, where \beta is the growth exponent. Ourresults support a generalization of the ansatz h = v t + (\Gamma t)^\beta \chi+ \eta + \zeta t^-\beta to higher dimensions, where v, \Gamma, \zeta and \etaare non-universal quantities whereas \beta and \chi are universal and the lastone depends on the surface geometry. Generalized Gumbel distributions providevery good fits of the distributions in at least four orders of magnitude aroundthe peak, which can be used for comparisons with experiments. Our numericalresults call for analytical approaches and experimental realizations of KPZclass in two-dimensional systems.

Kardar-Parisi-Zhang universality class in 2+1 dimensions: Universal geometry-dependent distributions and finite-time corrections