KP governs random growth off a one dimensional substrate

Abstract: The logarithmic derivative of the marginal distributions of randomly fluctuating interfaces in one dimension on a large scale evolve according to the Kadomtsev-Petviashvili (KP) equation. This is derived algebraically from a Fredholm determinant obtained in [MQR17, arXiv:1701.00018] for the KPZ fixed point as the limit of the transition probabilities of TASEP, a special solvable model in the KPZ universality class. The Tracy-Widom distributions appear as special self-similar solutions of KP and KdV. In addition, it is noted that several known exact solutions of the KPZ equation also solve KP.