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,Liu22] and the flat initial condition [Liu22] for the multipoint distributions, and the half-Brownian and Brownian initial conditions [JR22, Rah25] for the two-point distributions.

In this talk, I will talk about my recent work with Yuchen Liao in which we obtained 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 novelty is a probabilistic representation, inspired by [MQR21], of the kernel encoding the initial condition for TASEP, which was first defined through an implicit characterization in [Liu22]. Moreover, we also verify that the equal time degenerated version of our formula matches the path integral formula in [MQR21] for the KPZ fixed point.

This is a joint work with Yuchen Liao (University of Science and Technology of China).