From duality to determinants for q-TASEP and ASEP

We prove duality relations for two interacting particle systems: the $q$-deformed totally asymmetric simple exclusion process ($q$-TASEP) and the asymmetric simple exclusion process (ASEP). Expectations of the duality functionals correspond to certain joint moments of particle locations or integrated currents, respectively. Duality implies that they solve systems of ODEs. These systems are integrable and for particular step and half-stationary initial data we use a nested contour integral ansatz to provide explicit formulas for the systems' solutions, and hence also the moments. We form Laplace transform-like generating functions of these moments and via residue calculus we compute two different types of Fredholm determinant formulas for such generating functions. For ASEP, the first type of formula is new and readily lends itself to asymptotic analysis (as necessary to reprove GUE Tracy--Widom distribution fluctuations for ASEP), while the second type of formula is recognizable as closely related to Tracy and Widom's ASEP formula [Comm. Math. Phys. 279 (2008) 815--844, J. Stat. Phys. 132 (2008) 291--300, Comm. Math. Phys. 290 (2009) 129--154, J. Stat. Phys. 140 (2010) 619--634]. For $q$-TASEP, both formulas coincide with those computed via Borodin and Corwin's Macdonald processes [Probab. Theory Related Fields (2014) 158 225--400]. Both $q$-TASEP and ASEP have limit transitions to the free energy of the continuum directed polymer, the logarithm of the solution of the stochastic heat equation or the Hopf--Cole solution to the Kardar--Parisi--Zhang equation. Thus, $q$-TASEP and ASEP are integrable discretizations of these continuum objects; the systems of ODEs associated to their dualities are deformed discrete quantum delta Bose gases; and the procedure through which we pass from expectations of their duality functionals to characterizing generating functions is a rigorous version of the replica trick in physics.