An explicit formula of the Oslo stationary state

Abstract: In most sandpile models under driven-dissipative condition, a critical stationary state is known to emerge. This state displays organization features encoded in the dynamical (e.g. avalanche spectrum) and static (e.g. hyperuniformity) observables. However, the access to the explicit stationary state expression happens only in few cases, and for most sandpile models the question is still open. In this article, we derive such an expression for the Oslo model. To do so, we use different representations of the system configurations and of the dynamical process. Back and forth between these representations allows to tag invariant quantities to each configurations. The stationary probabilities are obtained by summing over all paths leading to a given configuration under the constraint specified by the invariants. This description corresponds exactly to a parametrized path integral formulation in discrete space-time.