| Abstract: |
| Let $\mathcal{G}$ be a graph equipped with random conductance. We give sufficient conditions for the Markov chain on $\mathcal{G}$ to exhibit large scale fluctuations in its on-diagonal heat kernel. The conditions also imply non-tightness of the height process w.r.t. the quenched measure, while the height process is tight w.r.t. the annealed measure. As an application of the general theory, we prove large scale fluctuations for the simple random walk on $1$-dimensional critical long-range percolation. This is based on a joint work with Zherui Fan and Takashi Kumagai. |
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