Combinatorial Properties of the Stationary States of Raise and Peel Models

In this work we consider the model of a stochastic process called «Raise-and-Peel» model. The model is explored using some algebraic techniques and the representation theory of the Temperley-Lieb algebra and the affine Hecke algebra. We introduce a special set of baxterized generators for the affine Hecke algebra and prove that they solve the Yang-Baxter and the reflection equations with spectral parameters. Then, using these baxterized generators and the relative trace operation we construct the so-called transfer-matrix, that is generating function for the commutative set in the affine Hecke algebra. The first derivative of the transfer matrix calculated for the value of its spectral parameter $x=1$ is then interpreted as an evolution operator of the stochastic Raise-and-Peel model. We study the model’s stationary state and observe some interesting combinatorial properties.