On the Martin boundary for discrete TASEP

We study a problem with three equivalent formulations: describing Gibbs measures for five-vertex model in quadrant; classifying coherent systems on a p-deformation of the Gelfand-Tsetlin graph related to Grothendieck polynomials; finding the Martin boundary for discrete time TASEP with p-geometric jumps. We find a wide family of the Gibbs measures, parameterized by certain analytic functions. A subset of our measures have probabilistic interpretation as interacting particle systems with fixed particles speeds. In contrast to previous related boundary problems, we find that admissible speeds are not arbitrary, but must be larger than $\frac{p}{1-p}$. For this subset we further establish Law of Large Numbers and Central Limit Theorem, connecting the fluctuations to families of independent GUE eigenvalues. As a consequence, the measures from the subset are extreme points of the Martin boundary. It remains open whether our list of measures is exhaustive.