Quasi-geodesics in integrable and non-integrable exclusion processes

Backwards geodesics for TASEP were introduced in [Fer18]. We consider flat initial conditions and show that under proper scaling its end-point converges to maximizer argument of the Airy${}_2$ process minus a parabola. We generalize its definition to generic non-integrable models including ASEP and speed changed ASEP (call it quasi-geodesics). We numerically verify that its end-point is universal, where the scaling coefficients are analytically computed through the KPZ scaling theory.