Avalanches in the Raise and Peel model in the presence of a wall

We investigate a non-equilibrium one-dimensional model known as the raise and peel model describing a growing surface which grows locally and has non-local desorption. For specific values of adsorption (u(a)) and desorption (u(d)) rates, the model shows interesting features. At u(a) = u(d), the model is described by a conformal field theory (with conformal charge c = 0) and its stationary probability can be mapped onto the ground state of the XXZ quantum chain. Moreover, for the regime u(a) >= u(d), the model shows a phase in which the avalanche distribution is scale-invariant. In this work, we study the surface dynamics by looking at avalanche distributions using a finite-sized scaling formalism and explore the effect of adding a wall to the model. The model shows the same universality for the cases with and without a wall for an odd number of tiles removed, but we find a new exponent in the presence of a wall for an even number of tiles released in an avalanche. New insights into the effect of parity on avalanche distributions are discussed and we provide a new conjecture for the probability distribution of avalanches with a wall obtained by using an exact diagonalization of small lattices and Monte Carlo simulations.