Scaling limits for non-convex interaction

The KPZ fixed point is conjectured to be the universal space-time scaling limit of the models belonging to the KPZ universality class and it was rigorously constructed by Matetski, Quastel and Remenik (Acta Math., 2021) as a scaling limit of TASEP (Totally Asymmetric Simple Exclusion Process) with arbitrary initial configuration. We set up a new, alternative approach to the KPZ fixed point, based on combinatorial structures and non-intersecting path constructions, which also allows studying inhomogeneous interacting particle systems. More specifically, we consider a discrete-time TASEP, where each particle jumps according to Bernoulli random variables with particle-dependent and time-inhomogeneous parameters, starting from an arbitrary initial configuration. We provide an explicit, step-by-step route from the very definition of the model to a Fredholm determinant representation of the joint distribution of the particle positions in terms of certain random walk hitting probabilities. Our tools include the combinatorics of the Robinson-Schensted-Knuth correspondence, intertwining relations, non-intersecting lattice paths, and determinantal point processes.