We establish general estimates for simple random walk on an arbitrary infinite random graph, assuming suitable bounds on volume and effective resistance for the graph. These are generalizations of the results in [6, Section 1,2], and in particular, imply the spectral dimension of the random graph. We will also give an application of the results to random walk on a long range percolation cluster. Key Words: Random walk -Random media -Heat kernel estimates -Spectral dimension -Long range percolation Running Head: Heat kernel estimates on random mediaRecently, there are intensive study for detailed properties of random walk on a percolation cluster. For a random walk on a supercritical percolation cluster on Z d (d ≥ 2), detailed Gaussian heat kernel estimates and quenched invariance principle have been obtained ([4, 10, 17, 19]). This means, such a random walk behaves in a diffusive fashion similar to a random walk on Z d . On the other hand, it is generally believed that random walk on a large critical cluster behaves subdiffusively (see [3] and the references therein). Critical percolation clusters are believed (and for some cases proved) to be finite. So, it is natural to consider random walk on an incipient infinite cluster (IIC), namely a critical percolation cluster conditioned to be infinite. Random walk on IICs has been proved to be subdiffusive on Z 2 ([13]), on trees ([14, 7]), and for the spread-out oriented percolation on Z d × Z + in dimension d > 6 ([6]).In order to study detailed properties of the random walk, it is nice and useful if one can compute the long time behaviour of the transition density (heat kernel). Let p n (x, y) be its transition density

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We consider random walks on random graphs determined by a some kind of continuum percolation on R. The vertex set of the random graph is given by the Poisson points conditioned that all points of Z are contained. The edge set of the random graph is determined by the random radii of the spheres centered at each points. We give heat kernel estimates for the random walks under the condition on the moment of the random radii. We will also discuss random walks on continuum percolation clusters in R d , d ≥ 2.