Sharpness of the percolation transition in the two-dimensional contact process

For ordinary (independent) percolation on a large class of lattices it is well known that below the critical percolation parameter $p_c$ the cluster size distribution has exponential decay and that power-law behavior of this distribution can only occur at $p_c$. This behavior is often called ``sharpness of the percolation transition.'' For theoretical reasons, as well as motivated by applied research, there is an increasing interest in percolation models with (weak) dependencies. For instance, biologists and agricultural researchers have used (stationary distributions of) certain two-dimensional contact-like processes to model vegetation patterns in an arid landscape (see [20]). In that context occupied clusters are interpreted as patches of vegetation. For some of these models it is reported in [20] that computer simulations indicate power-law behavior in some interval of positive length of a model parameter. This would mean that in these models the percolation transition is not sharp. This motivated us to investigate similar questions for the ordinary (``basic'') $2D$ contact process with parameter $\lambda$. We show, using techniques from Bollob\'{a}s and Riordan [8, 11], that for the upper invariant measure ${\bar{\nu}}_{\lambda}$ of this process the percolation transition is sharp. If $\lambda$ is such that (${\bar{\nu}}_{\lambda}$-a.s.) there are no infinite clusters, then for all parameter values below $\lambda$ the cluster-size distribution has exponential decay.