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We prove duality relations for two interacting particle systems: the$q$-deformed totally asymmetric simple exclusion process ($q$-TASEP) and theasymmetric simple exclusion process (ASEP). Expectations of the dualityfunctionals correspond to certain joint moments of particle locations orintegrated currents, respectively. Duality implies that they solve systems ofODEs. These systems are integrable and for particular step and half-stationaryinitial data we use a nested contour integral ansatz to provide explicitformulas for the systems' solutions, and hence also the moments. We formLaplace transform-like generating functions of these moments and via residuecalculus we compute two different types of Fredholm determinant formulas forsuch generating functions. For ASEP, the first type of formula is new andreadily lends itself to asymptotic analysis (as necessary to reprove GUETracy--Widom distribution fluctuations for ASEP), while the second type offormula 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'sMacdonald processes [Probab. Theory Related Fields (2014) 158 225--400]. Both$q$-TASEP and ASEP have limit transitions to the free energy of the continuumdirected polymer, the logarithm of the solution of the stochastic heat equationor 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 quantumdelta Bose gases; and the procedure through which we pass from expectations oftheir duality functionals to characterizing generating functions is a rigorousversion of the replica trick in physics.

From duality to determinants for q-TASEP and ASEP