Geodesic
KP governs random growth off a 1-dimensional substrate
Forum of Mathematics, Pi, Tome 10 (2022)

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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] for the Kardar–Parisi–Zhang (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 the KP and Korteweg–de Vries equations. In addition, it is noted that several known exact solutions of the KPZ equation also solve the KP equation. 
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Jeremy Quastel; Daniel Remenik. KP governs random growth off a 1-dimensional substrate. Forum of Mathematics, Pi, Tome 10 (2022). doi: 10.1017/fmp.2021.9

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