For TASEP, the backward geodesic endpoint converges to the argmax of Airy_2 minus a parabola, and the same universal law is conjectured and numerically supported for ASEP and speed-changed ASEP.
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Quasi-geodesics in integrable and non-integrable exclusion processes
For TASEP, the backward geodesic endpoint converges to the argmax of Airy_2 minus a parabola, and the same universal law is conjectured and numerically supported for ASEP and speed-changed ASEP.