Limits of Random Motzkin Paths with KPZ Related Asymptotics

Abstract We study Motzkin paths of length $L$ with general weights on the edges and endpoints. We investigate the limit behavior of the initial and final segments of the random Motzkin path viewed as a pair of processes starting from each of the two endpoints as $L$ becomes large. We then study macroscopic limits of the resulting processes, where in two different regimes we obtain Markov processes that appeared in the description of the stationary measure for the KPZ equation on the half-line and of conjectural stationary measure of the hypothetical KPZ fixed point on the half-line. Our results rely on the behavior of the Al-Salam–Chihara polynomials in the neighbourhood of the right end of their orthogonality interval and on the limiting properties of the $q$-Pochhammer and $q$-Gamma functions as $q\nearrow 1$.