From long random Motzkin paths to KPZ related asymptotics

We will consider Motzkin paths of length L with general weights on the edges and endpoints and discuss limiting  behavior of their initial and final segments as L becomes large. We then go for macroscopic limits of the resulting processes. In two different regimes, we obtain Markov processes which consitute the non-Brownian part 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. The results rely on the behavior of the Al-Salam–Chihara polynomials in the neighbourhood of the right end of the support  of their orthogonality measure and on Ramanujan-type limiting properties of the q-Pochhammer and q-Gamma functions as q->1.

This is a joint work with Wlodek Bryc (Univ. of Cincinnati) and Alexey Kuznetsov (York Univ. Toronto).