The polynuclear growth model (PNG) is a model for crystal growth in one dimension. It is one of the most basic models in the KPZ universality class, and in the droplet geometry it can be recast in terms of a Poissonized version of the longest increasing subsequence problem for a uniformly random permutation. In this talk I will explain a proof of a Fredholm determinant formula for the multipoint distributions of PNG with arbitrary initial data which is largely based on probabilistic arguments, relying on the invariant measure of the process and a time reversal symmetry property. This formula leads to a connection with the 2D Toda lattice.
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