All roads to the KPZ universality class

    On finite trees or graphs, ASEP is an irreducible Markov chain with a unique stationary distribution. Unlike the one-dimensional case, no simple product form generally exists. In infinite random trees, a nonzero flow can persist if branch capacities exceed the injection rate at the root; otherwise, the system becomes fully jammed [MR4355680].

Classical one-dimensional ASEP maps to the inviscid Burgers equation in the hydrodynamic limit. On general graphs, each edge follows a Burgers-type PDE, subject to coupling constraints at vertices (junctions). Simple merges (e.g., 2-to-1 lanes) already exhibit nontrivial shock and rarefaction waves, influencing phase diagrams and throughput.

While large deviation principles for current fluctuations are well-understood in 1D, explicit formulas on general graphs remain scarce. Macroscopic fluctuation theory suggests bottlenecks dominate rare-event statistics, and phase separation can occur if certain edges saturate.

Road networks with on- and off-ramps, intersections, or multi-lane traffic can be modeled by ASEP on graphs. Key phenomena include traffic jams localized around bottlenecks. In intracellular transport, motor proteins on complex filament networks create dense or jammed regions. Mathematical models align with observations of phase heterogeneity and spontaneous symmetry breaking [arXiv:1304.1943], [arXiv:1507.06166].