  We prove $\sqrt{\log n}$ lower bounds on the order of growth fluctuations in
three planar growth models (first-passage percolation, last-passage
percolation, and directed polymers) under no assumptions on the distribution of
vertex or edge weights other than the minimum conditions required for avoiding
pathologies. Such bounds were previously known only for certain restrictive
classes of distributions.In addition, the first-passage shape fluctuation
exponent is shown to be at least $1/8$, extending previous results to more
general distributions.
