For weighted Motzkin paths of length L we analyze the limit behavior, as L tends to infinity, of their initial and final segments. Macroscopic limits of the resulting stochastic processes, under two different regimes, are to Markov processes that earlier appeared in the description of
(1) the stationary measure for the KPZ equation on the half line;
(2) the conjectural stationary measure of the hypothetical KPZ fixed point on the half line.
On the technical side the results rely on behavior of the Al-Salam–Chihara polynomials in the neighbourhood of the upper end of their orthogonality interval and on the limiting properties of the q-Pochhammer and q-Gamma functions as q tends to 1. The talk is based on a joint work with Wlodek Bryc (Univ. of Cincinnati) and Alexei Kuznetsov (York Univ., Toronto).
