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It is shown that the polynuclear growth model is a completely integrableMarkov process in the sense that its transition probabilities are given byFredholm determinants of kernels produced by a scattering transform based onthe invariant measures modulo the absolute height, continuous time simplerandom walks. From the linear evolution of the kernels, it is shown that the$n$-point distributions are determinants of $n\times n$ matrices evolvingaccording to the two dimensional non-Abelian Toda lattice.

Polynuclear growth and the Toda lattice