The central limit theorem is one of the most important results in probability, and it has many generalizations in random processes and other stochastic models. In the last decades, the heat kernel estimate on percolation cluster and other random conductance models have been largely studied and many results of CLT type are obtained. In this talk, I will review these results and present a new heat kernel estimate, which can be seen as a quantitative local central limit theorem on the infinite percolation cluster. This work is a collaboration with Paul Dario and is published in Annals of Probability.

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