Abstract

Bond percolations on Cayley graphs are probabilistic processes in which edges are deleted randomly in an invariant way. In this talk, we will consider the question about the existence of a unique infinite cluster, which has deep links to geometry, group theory and ergodic theory. We first show a probabilistic characterization of Kazhdan’s property (T) which roughly speaking asserts that dense bond percolations form a unique infinite cluster. A complimentary result that Kazhdan groups admit some sparse bond percolation with a unique infinite cluster was proved in a recent breakthrough of Hutchcroft and Pete as the central ingredient in establishing that these groups have cost $1$. Motivated by the fixed price conjecture, the question whether such models can be built as a factor of i.i.d. was posed implicitly there and explicitly by Pete and Rokob. We show that the answer is affirmative for co-compact lattices in connected higher rank semisimple real Lie groups with property (T). Our proof uses a new phenomenon in continuum percolation on the associated symmetric space, which builds on recent breakthrough results about Poisson—Voronoi tessellations by Fraczyk, Mellick and Wilkens.   Based on joint works with Chiranjib Mukherjee (Munster) and with Jan Grebik (Leipzig).