  We study an oriented first passage percolation model for the evolution of a
river delta. This model is exactly solvable and occurs as the low temperature
limit of the beta random walk in random environment. We analyze the asymptotics
of an exact formula from [4] to show that, at any fixed positive time, the
width of a river delta of length $L$ approaches a constant times $L^{2/3}$ with
Tracy-Widom GUE fluctuations of order $L^{4/9}$. This result can be rephrased
in terms of particle systems. We introduce an exactly solvable particle system
on the integer half line and show that after running the system for only finite
time the particle positions have Tracy-Widom fluctuations.
