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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 JR21 [arXiv:1906.01053], Liu22 [arXiv:1907.09876] and the flat initial condition Liu22 [arXiv:1907.09876] for the multipoint distributions, and the half-Brownian and Brownian initial conditions JR22 [arXiv:2010.07357v1], Rah25 [arXiv:2504.19975]. 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. We also verify that the equal time degenerated version of our formula matches the path integral formula in MQR21 [arXiv:1701.00018] for the KPZ fixed point. 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 novelty is a probabilistic representation of the kernel encoding the initial condition for TASEP, which was first defined through an implicit characterization in Liu22 [arXiv:1907.09876].