SIMULTANEOUS UNIQ(H)UENESS OF INFINITE CLUSTERS IN STATIONARY RANDOM LABELED GRAPHS

In processes such as invasion percolation and certain models of contin uum percolation, in which a possibly random label f(b) is attached to each bond b of a possibly random graph, percolation models for various values of a parameter r are naturally coupled: one can define a bond b to be occupied at level r if f(b) less than or equal to r. If the la beled graph is stationary, then under the mild additional assumption o f positive finite energy, a result of Gandolfi, Keane, and Newman ensu res that, in lattice models, for each fixed r at which percolation occ urs, the infinite cluster is unique a.s. Analogous results exist for c ertain continuum models. A unifying framework is given for such fixed- r results, and it is shown that if the site density is finite and the labeled graph has positive finite energy, then with probability one, u niqueness holds simultaneously for all values of r. An example is give n to show that when the site density is infinite, positive finite ener gy does not ensure uniqueness, even for fixed r. In addition, with fin ite site density but without positive finite energy, one can have fixe d-r uniqueness a.s. for each r, yet not have simultaneous uniqueness.