IS24: Random media and limit theorems

Random walk approximation for the stationary distribution of the open ASEP

Dominik Schmid (University of Augsburg)

We investigate the structure of the stationary distribution of the open asymmetric simple exclusion process. More precisely, we show that the stationary distribution can be well-approximated by random walk measures. Our arguments rely on a recent characterization by Bryc of the stationary distribution as a two-layer Gibbs measure. Based on joint work with Zongrui Yang.

Domino tilings of Aztec diamond in random environment

Alexey Bufetov (Leipzig University)

It is well-known that under a certain choice of weights random domino tilings of the Aztec diamond can be analyzed via Schur measures on partitions. In this talk we will discuss the model in which the parameters of the Schur measure ( or, equivalently, some weights of dominoes) are random themselves. We establish the limit shape and global fluctuations results via the technique of Schur generating functions.

Random Motzkin paths with KPZ related asymptotics

Jacek Wesołowski (Politechnika Warszawska)

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 \(q\nearrow 1\).

Bibliography

\([1]\) Włodzimierz Bryc, Alexey Kuznetsov, Jacek Wesołowski (2025). "Limits of random Motzkin paths with KPZ related asymptotics." International Mathematics Research Notices, 4, 1-33.