  We propose a one-dimensional nonlocal stochastic model of adsorption and
desorption depending on one parameter, the adsorption rate. At a special value
of this parameter, the model has some interesting features. For example, the
spectrum is given by conformal field theory, and the stationary non-equilibrium
probability distribution is given by the two-dimensional equilibrium
distribution of the ice model with domain wall type boundary conditions. This
connection is used to find exact analytic expressions for several quantities of
the stochastic model. Vice versa, some understanding of the ice model with
domain wall type boundary conditions can be obtained by the study of the
stochastic model. At the special point we study several properties of the
model, such as the height fluctuations as well as cluster and avalanche
distributions. The latter has a long tail which shows that the model exhibits
self organized criticality. We also find in the stationary state a special
phase transition without enhancement and with a crossover exponent $\phi=2/3$.
Furthermore, we study the phase diagram of the model as a function of the
adsorption rate and find two massive phases and a scale invariant phase where
conformal invariance is broken.
