Under typical scaling, the last passage time field of the directed last passage percolation model with exponential site distributions converges to the KPZ fixed point. In this talk, we consider an atypical scenario in which the last passage time to a specific site is unusually large, and we explore how the last passage time field changes under this one-point upper large deviation event. We study a conditional law of large numbers and compute the limiting fluctuations in certain regimes. Our proofs rely on an analysis of explicit multi-point distributions. This talk is based on joint work with Jinho Baik and Dylan Cordaro.

In this talk I will introduce the multispecies TASEP with long-range swap, a new interacting particle system in which local backward-push and forward-jump rules combine to produce effective long-range exchanges. Despite its hybrid structure, the model is integrable: it satisfies two-particle reducibility, its two-species scattering matrix obeys the Yang–Baxter equation, and the Bethe ansatz yields explicit contour-integral formulas for transition probabilities. I will compare the model with both classical multispecies TASEP and long-range push dynamics, emphasizing the algebraic mechanisms that generate long-range swap behavior. I will also describe ongoing work on a broader interpolating family between TASEP-type swap and push-type swap rules, where integrability appears to persist only for certain parameter ranges. As a direction for future collaboration, I will briefly mention the problem of asymptotics for second-class particles in this long-range swap setting.