Comparing the number of infected vertices in two symmetric sets for Bernoulli percolation (and other random partitions)

For Bernoulli percolation on a given graph G = (V,E) we consider the cluster of some fixed vertex o \in V. We aim at comparing the number of vertices of this cluster in the set V_+ and in the set V_-, where V_+,V_- \subset V have the same size. Intuitively, if V_- is further away from o than V_+, it should contain fewer vertices of the cluster. We prove such a result in terms of stochastic domination, provided that o \in V_+, and V_+,V_- satisfy some strong symmetry conditions, and we give applications of this result in case G is a bunkbed graph, a layered graph, the 2D square lattice or a hypercube graph. Our result only relies on general probabilistic techniques and a combinatorial result on group actions, and thus extends to fairly general random partitions, e.g. as induced by Bernoulli site percolation or the random cluster model.