  We discuss a one-dimensional model of a fluctuating interface with a dynamic
exponent $z=1$. The events that occur are adsorption, which is local, and
desorption which is non-local and may take place over regions of the order of
the system size. In the thermodynamic limit, the time dependence of the system
is given by characters of the $c=0$ conformal field theory of percolation. This
implies in a rigorous way a connection between CFT and stochastic processes.
The finite-size scaling behavior of the average height, interface width and
other observables are obtained. The avalanches produced during desorption are
analyzed and we show that the probability distribution of the avalanche sizes
obeys finite-size scaling with new critical exponents.
