Quasi-Geodesics in Integrable and Non-Integrable Exclusion Processes

Abstract Backwards geodesics for TASEP were introduced in [30]. We consider flat initial conditions and show that under proper scaling the end-point of the geodesic converges to maximizer argument of the $$\hbox {Airy}_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mtext>Airy</mml:mtext> <mml:mn>2</mml:mn> </mml:msub> </mml:math> 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.