Spectral dimension of simple random walk on long-range random media

We give a general method that implies various on-diagonal heat kernel bounds, in particular, the lower bounds on stationary random media that may have long-range correlations. As an application, we determine the spectral dimension of simple random walk on the long-range percolation model in $Z^d$.

This is joint work with Van Hao Can (Hanoi) and David A. Croydon (Kyoto).