Stochastic growth in time-dependent environments

We study the Kardar-Parisi-Zhang (KPZ) growth equation in one dimension with a noise variance c(t) depending on time. We find that for c(t) proportional to t(-alpha) there is a transition at alpha = 1/2. When alpha > 1/2, the solution saturates at large times towards a nonuniversal limiting distribution. When alpha < 1/2 the fluctuation field is governed by scaling exponents depending on a and the limiting statistics are similar to the case when c(t) is constant. We investigate this problem using different methods: (1) Elementary changes of variables mapping the time-dependent case to variants of the KPZ equation with constant variance of the noise but in a deformed potential. (2) An exactly solvable discretization, the log-gamma polymer model. (3) Numerical simulations.