Supercritical loop percolation on Zd for d≥3

We consider supercritical percolation on Zd (d≥3) induced by random walk loop soup. Two vertices are in the same cluster if they are connected through a sequence of intersecting loops. We obtain quenched parabolic Harnack inequalities, Gaussian heat kernel bounds, the invariance principle and the local central limit theorem for the simple random walks on the unique infinite cluster. We also show that the diameter of finite clusters have exponential tails like in Bernoulli bond percolation. Our results hold for all d≥3 and all supercritical intensities despite polynomial decay of correlations.