Simple random walk on long range percolation clusters I: heat kernel bounds

In this paper, we derive upper bounds for the heat kernel of the simple random walk on the infinite cluster of a supercritical long range percolation process. For any d ≥ 1 and for any exponent \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${s \in (d, (d+2) \wedge 2d)}$$\end{document} giving the rate of decay of the percolation process, we show that the return probability decays like \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${t^{-{d}/_{s-d}}}$$\end{document} up to logarithmic corrections, where t denotes the time the walk is run. Our methods also yield generalized bounds on the spectral gap of the dynamics and on the diameter of the largest component in a box. The bounds and accompanying understanding of the geometry of the cluster play a crucial role in the companion paper (Crawford and Sly in Simple randomwalk on long range percolation clusters II: scaling limit, 2010) where we establish the scaling limit of the random walk to be α-stable Lévy motion.