Abstract: We consider Bernoulli percolation on transitive graphs of polynomial growth. In the subcritical regime (), it is well known that the connection probabilities decay exponentially fast.
In this talk, we discuss the supercritical phase , where we prove the exponential decay of the truncated connection probabilities (probabilities that two points are connected by an open path, but not to infinity). This sharpness result was established by Chayes, Chayes and Newman on and uses the difficult slab result of Grimmett and Marstrand. However the techniques used there are very specific to the hypercubic lattices and do not extend to more general geometries. Our approach involves new robust techniques based on the recent progress in the theory of sharp thresholds and the sprinkling method of Benjamini and Tassion. In this talk, we will mainly discuss the methods on , and give a completely new proof of the slab result of Grimmett and Marstrand.
Based on a joint work with Vincent Tassion.