The KPZ universality class governs fluctuations in a vast range of growth processes — from bacterial colonies to flame fronts to traffic flow. The KPZ fixed point, established for systems on the infinite line, characterizes the limiting distribution of height fluctuations at large times. The periodic case — growth on a ring — behaves differently because finite-size effects enforce recurrence: the growing interface eventually feels the topology of its domain. Establishing the periodic KPZ fixed point with general initial conditions requires controlling how arbitrary upper-semicontinuous starting profiles evolve through the totally asymmetric simple exclusion process at the relaxation time scale.

The key innovation is a representation of both the energy function and characteristic function through hitting expectation formulas — a probabilistic reformulation that converts the asymptotic analysis into a problem about random walks encountering barriers. This reformulation works because periodicity transforms the infinite-dimensional evolution into a compact setting where hitting probabilities carry all the necessary information.

The deeper insight: universality does not require infinite space. Confining a system to a periodic domain changes the fluctuation statistics but not the universality — the fixed point persists, modified by topology. The container alters the statistics without breaking the universality class, revealing that KPZ behavior is robust not just to microscopic details but to the global geometry of the space.

(arXiv:2603.01964)