Pinched-up periodic KPZ fixed point

The periodic KPZ fixed point is the conjectural universal limit of the KPZ universality class models on a ring when both the period and time critically tend to infinity. For the case of the periodic narrow wedge initial condition, we consider the conditional distribution when the periodic KPZ fixed point is unusually large at a particular position and time. We prove a conditional limit theorem up to the ``pinch-up" time. When the period is large enough, the result is the same as that for the KPZ fixed point on the line obtained by Liu and Wang in 2022. We identify the regimes in which the result changes and find probabilistic descriptions of the limits.