Multipoint distributions of the KPZ fixed point with compactly supported initial conditions

Abstract

The KPZ fixed point is a universal limiting space-time random field for the Kardar-Parisi-Zhang universality class. While the joint law of the KPZ fixed point at a fixed time has been studied extensively, the multipoint distributions of the KPZ fixed point in the general space-time plane are much less well understood. More explicitly, formulas were only available for the narrow wedge initial condition arXiv:1906.01053, arXiv:1907.09876 and the flat initial condition arXiv:1907.09876 for the multipoint distributions, and the half-Brownian and Brownian initial conditions arXiv:2010.07357v1, arXiv:2504.19975 for the two-point distributions. In this paper, we obtain the first formula for the space-time joint distributions of the KPZ fixed point with general initial conditions of compact support. The formula is obtained through taking $1:2:3$ KPZ scaling limit of the multipoint distribution formulas for the totally asymmetric simple exclusion process (TASEP). A key ingredient is a probabilistic representation, inspired by arXiv:1701.00018, of the kernel encoding the initial condition for TASEP, which was first defined through an implicit characterization in arXiv:1907.09876. Moreover, we also verify that the equal time version of our formula matches the path integral formula in arXiv:1701.00018 for the KPZ fixed point when the initial condition is of compact support.