Joint $q$-moments and shift invariance for the multi-species $q$-TAZRP\n on the infinite line

This paper presents a novel method for computing certain particle locations\nin the multi-species $q$-TAZRP (totally asymmetric zero range process). The\nmethod is based on a decomposition of the process into its discrete-time\nembedded Markov chain, which is described more generally as a monotone process\non a graded partially ordered set; and an independent family of exponential\nrandom variables. A further ingredient is explicit contour integral formulas\nfor the transition probabilities of the $q$-TAZRP. The main result of this\nmethod is a shift invariance for the multi-species $q$-TAZRP on the infinite\nline.\n By a previously known Markov duality result, these particle locations are the\nsame as joint $q$-moments. One particular special case is that for step initial\nconditions, ordered multi-point joint $q$-moments of the $n$-species $q$-TAZRP\nmatch the $n$-point joint $q$-moments of the single-species $q$-TAZRP. Thus, we\nconjecture that the Airy$_2$ process describes the joint multi-point\nfluctuations of multi-species $q$-TAZRP.\n As a probabilistic application of this result, we find explicit contour\nintegral formulas for the joint $q$-moments of the multi-species $q$-TAZRP in\nthe diffusive scaling regime.\n