2026/03/02 by Jinho Baik, Yuchen Liao, Zhipeng Liu
#math.PR #math-ph #math.MP
We consider the relaxation-time-scale limit of the periodic totally asymmetric simple exclusion process (PTASEP) with general initial conditions. For every sequence of initial conditions approximating a periodic upper semicontinuous function, we compute the limiting space-time multipoint distributions of the rescaled particle locations and height functions. The resulting finite-dimensional distributions are explicit and form a consistent family, thereby defining a spatially periodic space-time random field. We call this field the periodic KPZ fixed point with the corresponding initial condition. This extends earlier results for PTASEP with special initial conditions and defines the periodic analogue of the KPZ fixed point on the line. The main technical novelty is a pair of new probabilistic representations for the energy function and the characteristic function, the two functions through which the initial condition enters the finite-time PTASEP multipoint distribution formula. Both representations are expressed in terms of a geometric random walk and two stopping times, namely the first hitting time of the initial profile and the first such hitting time at or after one full period, with the latter capturing the periodic geometry.