Isoperimetry and heat kernel decay on percolation clusters

We prove that the heat kernel on the infinite Bernoulli percolation cluster in ᵈ almost surely decays faster than t-d/2. We also derive estimates on the mixing time for the random walk confined to a finite box. Our approach is based on local isoperimetric inequalities. Some of the results of this paper were previously announced in the note of Mathieu and Remy [C. R. Acad. Sci. Paris Sér. I Math. 332 (2001) 927--931].